Un Exemple Détaillé

Un Exemple Détaillé _________

Quelques Exemples Réduits

Un Exemple Détaillé

Les Coefficients Sobaliens {GFG}

Des Triplets Pythagoriciens

L'Hexagone et Les Cubes Magiques

* * *

Un Exemple Détaillé

'____________________________________________________

' Les sommes des puissances des coefficients binomiaux

' des diagonales principales du Triangle de Pascal

'_________________________

' mathématiques récréatives

'_________________________

'_______

' Exemple___________________________________________________________

'_________________________1^5 +3^5 +6^5 +10^5 +15^5 +21^5 +28^5 ...

'__________________________________________________________________

' Génération de la formule de la somme des puissances 5ièmes {p=5}

' des coefficients binomiaux de la 3ième diagonale principale {q=2}

' du Triangle de Pascal

' { i.e. "suite des nombres triangulaires" }

'__________________________________________________________________

'____________________________________________________________________________

'__i=n___C(i+1,2)______Somme_{i=1..n} C(i+1,2)^5_____________________>>>

'___1______1_________1^5_________________________________->______________1

'___2______3_________1^5+3^5____________________________->____________244

'___3______6_________1^5+3^5+6^5_______________________->___________8020

'___4_____10_________1^5+3^5+6^5+10^5_________________->_________108020

'___5_____15_________1^5+3^5+6^5+10^5+15^5____________->_________867395

'___6_____21_________1^5+3^5+6^5+10^5+15^5+21^5______->________4951496

'___7_____28_________....+6^5+10^5+15^5+21^5+28^5______->_______22161864

'___8_____36_________...+10^5+15^5+21^5+28^5+36^5______->_______82628040

'___9_____45_________...+15^5+21^5+28^5+36^5+45^5______->______267156165

'__10_____55_________...+21^5+28^5+36^5+45^5+55^5______->______770440540

'__11_____66_________...+28^5+36^5+45^5+55^5+66^5______->_____2022773116

'__12_____78_________...+36^5+45^5+55^5+66^5+78^5______->_____4909947484

'__13_____91_________...+45^5+55^5+66^5+78^5+91^5______->____11150268935

'____________________________________________________________________________

'________Somme_{i=1..n}_____________________________Différences_===>>>

'__i=n____C(i+1,2)^5_______C(n+2,3)______________D_n____{"Deltas"}____________>>>

'___1_______________1_________1________________1*77/1_____->____________77 '___2_____________244_________4______________244*77/4_____->__________4697 '___3____________8020________10____________8020*77/10_____->_________61754 '___4__________108020________20__________108020*77/20_____->________415877 '___5__________867395________35__________867395*77/35_____->_______1908269 '___6_________4951496________56_________4951496*77/56_____->_______6808307 '___7________22161864________84________22161864*77/84_____->______20315042 '___8________82628040_______120_______82628040*77/120_____->______53019659 '___9_______267156165_______165______267156165*77/165_____->_____124672877 '__10_______770440540_______220______770440540*77/220_____->_____269654189 '__11______2022773116_______286_____2022773116*77/286_____->_____544592762 '__12______4909947484_______364_____4909947484*77/364_____->____1038642737 '__13_____11150268935_______455____11150268935*77/455_____->____1886968589 '____________________________________________________________________________

'__Note_:__Le facteur "77" (="A") est purement arbitraire; c'est seulement

'__________le "plus petit multiple commun" ("PPMC") qui permettra de

'__________transformer en entiers naturels les coefficients du polynôme

'__________non-réduit qu'on trouvera ci-après. '____________________________________________________________________________

'____________________________________________________________________________ '__________________________Différences_===>>>_______Différences_===>>>

'__i=n____D_n____{"Deltas"}________________________________________>>>

'____________C(n-1,0) =1

'________________/__________C(n-1,1)

'___1_=>__________77___________/_________C(n-1,2)

'___2_=>________4697__________4620___________/_________C(n-1,3) '___3_=>_______61754_________57057_________52437___________/ '___4_=>______415877________354123________297066________244629____->________ '___5_=>_____1908269_______1492392_______1138269________841203____->________ '___6_=>_____6808307_______4900038_______3407646_______2269377____->________ '___7_=>____20315042______13506735_______8606697_______5199051____->________ '___8_=>____53019659______32704617______19197882______10591185____->________ '___9_=>___124672877______71653218______38948601______19750719____->________ '__10_=>___269654189_____144981312______73328094______34379493____->________ '__11_=>___544592762_____274938573_____129957261______56629167____->________ '__12_=>__1038642737_____494049975_____219111402______89154141____->________ '__13_=>__1886968589_____848325852_____354275877_____135164475____->________ '____________________________________________________________________________

'____________________________________________________________________________ '__________________________Différences_===>>>_______Différences_===>>>

'__i=1____D_n____{"Deltas"}________________________________________>>>

'___1________________LES SOMMES DES PUISSANCES DES COEFFICIENTS

'___2______________________BINOMIAUX DES DIAGONALES PRINCIPALES

'___3______C(n-1,4)________________________DU TRIANGLE DE PASCAL

'___4__________/_______C(n-1,5)____< mathématiques récréatives >

'___5_=>____596574________/_______C(n-1,6)

'___6_=>___1428174______831600________/______C(n-1,7)

'___7_=>___2929674_____1501500_____669900________/_____C(n-1,8)

'___8_=>___5392134_____2462460_____960960_____291060_______/ '___9_=>___9159534_____3767400____1304940_____343980_____52920 '__10_=>__14628774_____5469240____1701840_____396900_____52920 '__11_=>__22249674_____7620900____2151660_____449820_____52920 '__12_=>__32524974____10275300____2654400_____502740_____52920

'__13_=>__46010334____13485360____3210060_____555660_____52920

'____________________________________________________________________________

'******************** => Formule générée => .................................

'______________________________________________________________________ '____________________________________________polynôme non-réduit

'____________________________________________///////////////////////////

' Somme_{i=1..n} C(i+1,2)^5 = C(n+2,3) *

' {77 +4620*C(n-1,1) +52437*C(n-1,2) +244629*C(n-1,3) +596574*C(n-1,4)

' +831600*C(n-1,5) +669900*C(n-1,6) +291060*C(n-1,7) +52920*C(n-1,8)}/77

'______________ {"triangulaires"} -------------------

'______________________________________________________________________ '________________________________________________polynôme réduit '________________________________________________/////////////////////

' Somme_{i=1..n} [ ({(i+1)!}/{[2!]*[(i-1)!]})^5 ] =

' [ 113400*n^11 +1247400*n^10 +5544000*n^9 +12474000*n^8 +14196600*n^7

' +6237000*n^6 -831600*n^5 +1108800*n^3 -172800*n ]/11!

'___________________________________________________________

'>>>___>>>___>>> 1^5 +3^5 +6^5 +10^5 +15^5 +21^5 +28^5 ... OEIS A085440

'^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

'*****************************************************************************

-> Pour scruter une série du " Dictionnaire en ligne des suites numériques " { Encyclopédie en ligne des suites de

nombres entiers }

[ On-line Encyclopedia of Integer Sequences ],

veuillez procéder comme suit.

-> Entrez ci-après comme dans l'exemple proposé les nombres séquentiels

vous intéressant, puis cliquez sur l'onglet "Soumettre".

|*| Somme_{i= 1..n} C(i +1, 2)^p = 3 * C(n +2, 3) * Somme_{j= 1..2^p-1} a(j, p, 2) * C(n-1, j-1)/(j +2) <= grande formule générale combinatorielle

( pour q = 2, triangulaires )

|*| a(j, p, 2) = Somme_{k= 1..[2*j +1 +(-1)^(j-1)]/4} [ C(j-1, 2*k -2) * C(j +3 -2*k, j +1 -2*k)^(p-1) <= formule générale des coefficients sobaliens

|*| - C(j-1, 2*k -1) * C(j +2 -2*k, j -2*k)^(p-1) ] ( pour q = 2, triangulaires )

Exemple pour p = 5 :

-------------------------

[ coefficients sobaliens ]

a(1,5,2) = 1

a(2,5,2) = 3^4 -1 = 80

a(3,5,2) = 3^4*[2^4-2] +1 = 1135

a(4,5,2) = [5!/(3!2!)]^(5-1) -3*[4!/(2!2!)]^(5-1) +3*3^(5-1) -1 = 6354

a(5,5,2) = C(6,4)^4 -4*C(5,3)^4 +6*C(4,2)^4 -4*C(3,1)^4 +C(2,0)^4

= 15^4 -4*10^4 +6*6^4 -4*3^4 +1

= 50625 -40000 +7776 -324 +1 = 18078

a(6,5,2) = C(7,5)^4 -5*C(6,4)^4 +10*C(5,3)^4 -10*C(4,2)^4 +5*C(3,1)^4 -C(2,0)^4

= 21^4 -5*15^4 +10*10^4 -10*6^4 +5*3^4 -1

= 194481 -253125 +100000 -12960 +405 -1 = 28800

. . . . . . .

a(7,5,2) = C(8,6) -6*C(7,5)^4 +15*C(6,4)^4 -20*C(5,3)^4 +15*C(4,2)^4 -6*C(3,1)^4 +1*C(2,0)^4 = 26100

= [ (6*5^4 -20*5^3 +21*5^2 -7*5)*6! ]/[ 3*2^4 ] = 26100

. . . . . . .

a(8,5,2) = 5*8!/(2!)^4 =12600

= [ (5^2-5)*7! ]/2^3 = 12600

a(9,5,2) = 8!/(2!)^4 = 2520

= 4*7!/2^3 = 2520

Somme_{i=1..n} C(i +1, 2)^5 = 3 * [ C(n+2, 3) ] * [ 1/3 +(80/4)*C(n-1, 1) +(1135/5)*C(n-1, 2) +(6354/6)*C(n-1, 3)

+(18078/7)*C(n-1, 4) +(28800/8)*C(n-1, 5) +(26100/9)*C(n-1, 6)

+(12600/10)*C(n-1, 7) +(2520/11)*C(n-1,8) ]

Exemple pour n = 7 : ( voir OEIS - A085440 )

-------------------------

Somme_{i=1..7} C(i +1, 2)^5 = a(7) = 1^5 +3^5 +6^5 +10^5 +15^5 +21^5 +28^5

= 3 * [ C(9, 3)] * [ 1/3 +20*6 +227*15 +1059*20 +(18078/7)*15 +3600*6 +2900*1

+1250*0 +(2520/11)*0 ]

= 3 * [ 84 ] * [ 1/3 +... +2900 +0 +0 ]

= 84 +30240 +858060 +5337360 +9762120 +9762120 +5443200 +730800 +0 +0

= 22161864

OEIS -> A087127

1=a(1,1,2),

1=a(1,2,2), 2=a(2,2,2), 1=a(3,2,2),

1=a(1,3,2), 8=a(2,3,2), 19=a(3,3,2), 18=a(4,3,2), 6=a(5,3,2),

1, 26, 163, 432, 564, 360, 90=a(7,4,2),

1, 80, 1135, 6354, 18078, 28800, 26100, 12600, 2520=a(9,5,2),

1, 242, 7291, 77400, 405060, 1210680, 2211570, 2520000, 1751400, 680400, 113400 = a(11,6,2),

1, 728, 45199, 862218, 7667646, 38350080, 118848420, 239992200, 322176960, 285768000,

___161141400, 52390800, 7484400 = a(13,7,2),

1, 2186, 275563, 9166752, 132530244, 1044003240, 5053473810, 16100582400, 35105157360,

___53352885600, 56567775600, 41074387200, 19489377600, 5448643200, 681080400 = a(15,8,2),

1, 6560, 1666495, 94980834, 2172942078, 25991642880, 187333356420, 884539567800, 2880471280680,

___6673493836800, 11182022686800, 13605157288800, 11914610954400, 7322976460800, 2999478081600,

___735566832000, 81729648000 = a(17,9,2),

1, 19682, 10038331, 969825960, 34503746820, 611962571160, 6369577111890, . . . . ., 572516184240000,

___125046361440000, 12504636144000 = a(19,10,2),

1, a(2,11,2) = 59048 = 3^10-1, a(3,11,2) = 60348079, a(4,11,2) = 9818778618, . . . . ., a(19,11,2), a(20,11,2),

___a(21,11,2) = 2375880867360000,

1, a(2,12,2) =177146 = 3^11-1, a(3,12,2) = 362442763, a(4,12,2), . . . . ., a(21,12,2), a(22,12,2),

___ a(23,12,2) = 548828480360160000,

1, a(2,13,2), a(3,13,2), a(4,13,2), . . . . ., a(23,13,2), a(24,13,2), a(25,13,2),

1, . . . . .,


	=> ... LES COEFFICIENTS SOBALIENS {GFG}


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