A087127 -id:A087127 - cp's OEIS frontend

A086024 a(n) = Sum_{i=1..n} C(i+3,4)^3.

1, 126, 3501, 46376, 389376, 2389752, 11650752, 47587752, 168875127, 534401002, 1537404003, 4080706128, 10109274128, 23590546128, 52243162128, 110473767504, 224205418629, 438589465254, 830009446129, 1524339072504, 2724140666880, 4748425291880, 8089787666880

Offset: 1

André F. Labossière, Jul 11 2003

T. D. Noe, Table of n, a(n) for n = 1..1000
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 13.
Index entries for linear recurrences with constant coefficients, signature (14,-91,364,-1001,2002,-3003,3432, -3003,2002,-1001,364,-91,14,-1).

Cf. A087127, A024166, A085438 - A085442, A086020 - A086030.

[(n/69189120)*(13824 + 960960*n^2 + 5885880*n^3 + 14370356*n^4 + 19269250*n^5 + 15996695*n^6 + 8678670*n^7 + 3138135*n^8 + 750750*n^9 + 114205*n^10 + 10010*n^11 + 385*n^12): n in [1..30]]; // G. C. Greubel, Nov 22 2017

Table[(n/69189120)*(13824 + 960960*n^2 + 5885880*n^3 + 14370356*n^4 + 19269250*n^5 + 15996695*n^6 + 8678670*n^7 + 3138135*n^8 + 750750*n^9 + 114205*n^10 + 10010*n^11 + 385*n^12), {n,1,30}] (* G. C. Greubel, Nov 22 2017 *)
LinearRecurrence[{14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1},{1,126,3501,46376,389376,2389752,11650752,47587752,168875127,534401002,1537404003,4080706128,10109274128,23590546128},30] (* Harvey P. Dale, Feb 18 2024 *)

for(n=1,30, print1(sum(k=1,n, binomial(k+3, 4)^3), ", ")) \\ G. C. Greubel, Nov 22 2017

a(n) = ( C(n+4, 5)/1001 )*( 1001 +20020*C(n-1, 1) +125840*C(n-1, 2) +390390*C(n-1, 3) +695695*C(n-1, 4) +750750*C(n-1, 5) +486850*C(n-1, 6) +175175*C(n-1, 7) +26950*C(n-1, 8) ).
G.f.: x*(1 +112*x +1828*x^2 +8464*x^3 +13840*x^4 +8464*x^5 +1828*x^6 +112*x^7 +x^8)/(x-1)^14 . - R. J. Mathar, Dec 22 2013
-(n-1)^3*a(n) +2*(n+1)*(n^2+2*n+13)*a(n-1) -(n+3)^3*a(n-2)=0. - R. J. Mathar, Dec 22 2013
a(n) = (n/69189120)*(13824 + 960960*n^2 + 5885880*n^3 + 14370356*n^4 + 19269250*n^5 + 15996695*n^6 + 8678670*n^7 + 3138135*n^8 + 750750*n^9 + 114205*n^10 + 10010*n^11 + 385*n^12). - G. C. Greubel, Nov 22 2017

A086029 a(n) = Sum_{i=1..n} C(i+6,7)^2.

Original entry on oeis.org

1, 65, 1361, 15761, 124661, 751925, 3696581, 15475205, 56884430, 187758030, 565982734, 1578749710, 4117700254, 10127050654, 23648089054, 52733344990, 112835299639, 232623278455, 463695768455, 896396608455, 1684993889355, 3086944610955, 5522978819355

Offset: 1

André F. Labossière, Jul 11 2003

			a(3) = C(10,8)*(-1*C(10,0) + 8*C(10,1) - 36*C(10,2) + 120*C(10,3) - 330*C(10,4) + 792*C(10,5) - 1716*C(10,6) + 3432*C(10,7))/6435 = 1361.

G. C. Greubel, Table of n, a(n) for n = 1..5000
John Engbers and Christopher Stocker, Two Combinatorial Proofs of Identities Involving Sums of Powers of Binomial Coefficients, Integers 16 (2016), #A58.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 13.
Index entries for linear recurrences with constant coefficients, signature (16, -120, 560, -1820, 4368, -8008, 11440, -12870, 11440, -8008, 4368, -1820, 560, -120, 16, -1).

Cf. A087127, A024166, A085438-A085442, A086020-A086030, A234253.

[(1/108972864000)*n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7+n)*(7 + 2*n)*(60 + 64974*n + 126245*n^2 + 82467*n^3 + 23408*n^4 + 3003*n^5 + 143*n^6): n in [1..30]]; // G. C. Greubel, Nov 22 2017

A086029:=n->add(binomial(i+6, 7)^2, i=1..n); seq(A086029(n), n=1..30); # Wesley Ivan Hurt, Dec 22 2013

Table[Sum[Binomial[i + 6, 7]^2, {i, n}], {n, 30}] (* Wesley Ivan Hurt, Dec 22 2013 *)
LinearRecurrence[{16,-120,560,-1820,4368,-8008,11440,-12870,11440,-8008,4368,-1820,560,-120,16,-1},{1,65,1361,15761,124661,751925, 3696581, 15475205, 56884430,187758030,565982734,1578749710,4117700254, 10127050654,23648089054,52733344990},40] (* Harvey P. Dale, Apr 25 2016 *)

a(n) = sum(i=1, n, binomial(i+6, 7)^2); \\ Michel Marcus, Dec 22 2013

Vec(x*(x+1)*(x^6+48*x^5+393*x^4+832*x^3+393*x^2+48*x+1)/(x-1)^16 + O(x^100)) \\ Colin Barker, May 02 2014

a(n) = C(n+7,8)*(-C(n+7,0) + 8*C(n+7,1) - 36*C(n+7,2) + 120*C(n+3,7) - 330*C(n+7,4) + 792*C(n+7,5) - 1716*C(n+7,6) + 3432*C(n+7,7))/6435. - Yahia Kahloune, Dec 22 2013
(n-1)^2*a(n) +(-2*n^2-10*n-37)*a(n-1) +(n+6)^2*a(n-2)=0. - R. J. Mathar, Dec 22 2013
G.f.: x*(x+1)*(x^6 +48*x^5 +393*x^4 +832*x^3 +393*x^2 +48*x +1)/(1-x)^16. - Colin Barker, May 02 2014
a(n) = (1/108972864000)*n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7+n)*(7 + 2*n)*(60 + 64974*n + 126245*n^2 + 82467*n^3 + 23408*n^4 + 3003*n^5 + 143*n^6). - G. C. Greubel, Nov 22 2017

More terms from Michel Marcus, Dec 22 2013

A086021 a(n) = Sum_{i=1..n} C(i+2,3)^3.

Original entry on oeis.org

1, 65, 1065, 9065, 51940, 227556, 820260, 2548260, 7040385, 17688385, 41082041, 89310585, 183506960, 359122960, 673554960, 1216893456, 2126746665, 3608290665, 5960927665, 9613191665, 15167828676, 23459298500, 35626298500, 53202298500, 78227501625, 113386110201

Offset: 1

André F. Labossière, Jul 11 2003

G. C. Greubel, Table of n, a(n) for n = 1..5000
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 13.
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A086020, A086022, A086023, A086024, A086025, A086026, A086027, A086028, A086029, A086030, A087127, A024166, A085438, A085439, A085440, A085441, A085442.

[n^2*(-36 + 300*n + 1535*n^2 + 2700*n^3 + 2442*n^4 + 1260*n^5 + 375*n^6 + 60*n^7 + 4*n^8)/8640: n in [1..30]]; // G. C. Greubel, Nov 22 2017

Table[n^2*(-36 + 300*n + 1535*n^2 + 2700*n^3 + 2442*n^4 + 1260*n^5 + 375*n^6 + 60*n^7 + 4*n^8)/8640, {n, 1, 30}] (* G. C. Greubel, Nov 22 2017 *)

Vec(-x*(x^6+54*x^5+405*x^4+760*x^3+405*x^2+54*x+1)/(x-1)^11 + O(x^100)) \\ Colin Barker, May 02 2014

for(n=1,30, print1(n^2*(-36 + 300*n + 1535*n^2 + 2700*n^3 + 2442*n^4 + 1260*n^5 + 375*n^6 + 60*n^7 + 4*n^8)/8640, ", ")) \\ G. C. Greubel, Nov 22 2017

a(n) = (C(n+3, 4)/1)*(1 +12*C(n-1, 1) +46*C(n-1, 2) +84*C(n-1, 3) +81*C(n-1, 4) +40*C(n-1, 5) +8*C(n-1, 6)). - Edited by Colin Barker, May 02 2014
G.f.: -x*(x^6 +54*x^5 +405*x^4 +760*x^3 +405*x^2 +54*x +1) / (x-1)^11. - Colin Barker, May 02 2014
a(n) = n^2*(-36 + 300*n + 1535*n^2 + 2700*n^3 + 2442*n^4 + 1260*n^5 + 375*n^6 + 60*n^7 + 4*n^8)/8640. - G. C. Greubel, Nov 22 2017

A086026 a(n) = Sum_{i=1..n} C(i+4,5)^3.

Original entry on oeis.org

1, 217, 9478, 185094, 2185470, 18188478, 116799606, 613592694, 2745339597, 10769363605, 37850444632, 121189368664, 358136205336, 987118431768, 2559344776920, 6286103520984, 14712254089533, 32974344717237, 71073599975686, 147860902015750, 297836101312750

Offset: 1

André F. Labossière, Jul 11 2003

			a(3) = C(8,6)^2*(1 + 279*C(3,1) + 681*C(3,2) + 504*C(3,3))/280 = 9478. - _Yahia Kahloune_, Dec 22 2013

G. C. Greubel, Table of n, a(n) for n = 1..5000
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 13.
Index entries for linear recurrences with constant coefficients, signature (17,-136,680,-2380,6188,-12376, 19448,-24310,24310,-19448,12376,-6188,2380,-680,136,-17,1).

Cf. A087127, A024166, A085438 - A085442, A086020 - A086030.

[(n^2/580608000)*(57600 + 4583040*n + 28668304*n^2 + 80791200*n^3 + 133134680*n^4 + 142979760*n^5 + 105929613*n^6 + 55881000*n^7 + 21323540* n^8 + 5904360*n^9 + 1175062*n^10 + 163800*n^11 + 15180*n^12 + 840*n^13 + 21*n^14): n in [1..30]]; // G. C. Greubel, Nov 22 2017

A086026 := proc(n)
    add( binomial(i+4,5)^3,i=1..n) ;
end proc:
seq(A086026(n),n=1..30) ; # R. J. Mathar, Dec 22 2013

Table[Sum[Binomial[i + 4, 5]^3, {i, n}], {n, 30}] (* Wesley Ivan Hurt, Dec 22 2013 *)

a(n) = sum(i=1, n, binomial(i+4, 5)^3); \\ Michel Marcus, Dec 22 2013

a(n) = C(n+5,6)^2*(1 + 279*C(n,1) + 681*C(n,2) + 504*C(n,3) + 126*C(n,4) )/280. - Yahia Kahloune, Dec 22 2013
-(n-1)^3*a(n) +(2*n+3)*(n^2+3*n+21)*a(n-1) -(n+4)^3*a(n-2)=0. - R. J. Mathar, Dec 22 2013
G.f.: -x*(x^10 +200*x^9 +5925*x^8 +52800*x^7 +182700*x^6 +273504*x^5 +182700*x^4 +52800*x^3 +5925*x^2 +200*x +1) / (x -1)^17. - Colin Barker, May 02 2014
a(n) = (n^2/580608000)*(57600 + 4583040*n + 28668304*n^2 + 80791200*n^3 + 133134680*n^4 + 142979760*n^5 + 105929613*n^6 + 55881000*n^7 + 21323540* n^8 + 5904360*n^9 + 1175062*n^10 + 163800*n^11 + 15180*n^12 + 840*n^13 + 21*n^14). - G. C. Greubel, Nov 22 2017

More terms from Michel Marcus, Dec 22 2013

A086027 a(n) = Sum_{i=1..n} binomial(i+5,6)^2.

Original entry on oeis.org

1, 50, 834, 7890, 51990, 265434, 1119210, 4063866, 13081875, 38131900, 102259964, 255425340, 600047436, 1336192860, 2838530460, 5783112156, 11350211925, 21540508734, 39656591950, 71021001950, 124026854850, 211648774950, 353581802550, 579225802950, 931794553575

Offset: 1

André F. Labossière, Jul 11 2003

T. D. Noe, Table of n, a(n) for n = 1..1000
John Engbers and Christopher Stocker, Two Combinatorial Proofs of Identities Involving Sums of Powers of Binomial Coefficients, Integers 16 (2016), #A58.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 13..
Index entries for linear recurrences with constant coefficients, signature (14, -91, 364, -1001, 2002, -3003, 3432, -3003, 2002, -1001, 364, -91, 14, -1).

Cf. A087127, A024166, A085438, A085439, A085440, A085441, A085442, A086020, A086021, A086022, A086023, A086024, A086025, A086026, A086028, A086029, A086030.

List([1..30], n-> Sum([1..n], j-> Binomial(j+5,6)^2)); # G. C. Greubel, Aug 27 2019

[n*(n+1)*(n+2)*(n+3)*(n+4)*(n+5)*(n+6)*(77*n^6 +1386*n^5 +9380*n^4 + 29400*n^3 +41783*n^2 +20874*n +60)/518918400: n in [1..30]]; // G. C. Greubel, Nov 22 2017

A086027:=n->sum(binomial(i+5,6)^2, i=1..n); seq(A086027(k), k=1..50); # Wesley Ivan Hurt, Oct 24 2013

Table[Sum[Binomial[k + 5, 6]^2, {k, 1, n}], {n, 50}] (* Wesley Ivan Hurt, Oct 24 2013 *)

vector(30, n, sum(i=1,n, binomial(i+5,6)^2) ) \\ G. C. Greubel, Nov 22 2017

[sum(binomial(j+5,6)^2 for j in (1..n)) for n in (1..30)] # G. C. Greubel, Aug 27 2019

From R. J. Mathar, Jun 16 2010: (Start)
G.f.: x*(1 + 36*x + 225*x^2 + 400*x^3 + 225*x^4 + 36*x^5 + x^6)/(1-x)^14.
a(n) = n*(n+1)*(n+2)*(n+3)*(n+4)*(n+5)*(n+6)*(77*n^6 + 1386*n^5 + 9380*n^4 + 29400*n^3 + 41783*n^2 + 20874*n + 60)/518918400. (End)

More terms from R. J. Mathar, Jun 16 2010

A086028 a(n) = Sum_{i=1..n} C(i+5,6)^3.

Original entry on oeis.org

1, 344, 22296, 615000, 9876000, 108487128, 897376152, 5950405848, 33031486875, 158406862000, 671944398512, 2567519091888, 8965083682032, 28938181326000, 87168786702000, 246953567853744, 662331582918141, 1691011474896264, 4129363811437000, 9684000822437000

Offset: 1

André F. Labossière, Jul 11 2003

			a(4) = Sum_{i=1..4} C(i+5,6)^3 = C(6,6)^3 + C(7,6)^3 + C(8,6)^3 + C(9,6)^3 = 1^3 + 7^3 + 28^3 + 84^3 = 615000.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 13.
Index entries for linear recurrences with constant coefficients, signature (20, -190, 1140, -4845, 15504, -38760, 77520, -125970, 167960, -184756, 167960, -125970, 77520, -38760, 15504, -4845, 1140, -190, 20, -1).

Cf. A087127, A024166, A085438 - A085442, A086020, A086021 - A086030.

[(n/120679663104000)*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(-864000 + 2116800*n + 772737840*n^2 + 3398930472*n^3 + 6406454992 *n^4 + 6701566410*n^5 + 4302755765*n^6 + 1780394616*n^7 + 484074591*n^8 + 85975890*n^9 + 9604595*n^10 + 612612*n^11 + 17017*n^12): n in [1..30]]; // G. C. Greubel, Nov 22 2017

A086028 := proc(n)
    add( binomial(i+5,6)^3,i=1..n) ;
end proc:
seq(A086028(n),n=1..30) ; # R. J. Mathar, Dec 22 2013

Table[Sum[Binomial[k+5,6]^3, {k,1,n}], {n,1,30}] (* G. C. Greubel, Nov 22 2017 *)

for(n=1, 30, print1(sum(k=1,n, binomial(k+5,6)^3), ", ")) \\ G. C. Greubel, Nov 22 2017

-(n-1)^3*a(n) +2*(n+2)*(n^2 +4*n +31)*a(n-1) -(n+5)^3*a(n-2)=0. - R. J. Mathar, Dec 22 2013
From Yahia Kahloune, Dec 23 2013; (Start)
a(n) = C(n+6,7)*(-15*F6(n) + 63063*(7*C(n+11,12) + 195*C(n+10,12) + 920*C(n+9,12) + 920*C(n+8,12) + 195*C(n+7,12) + 7*C(n+6,12)))/415701;
where F6(n) = Sum_{i=0..6} (-1)^i*C(6+i,i)*C(n+6,i) = C(6,0)*C(n+6,0) - C(7,1)*C(n+6,1) + C(8,2)*C(n+6,2) - C(9,3)*C(n+6,3) + C(10,4)*C(n+6,4) - C(11,5)*C(n+6,5) + C(12,6)*C(n+6,6).
The values of F6(n), (n=0...9) are: 1, 1716, 10725, 39754, 112827, 270348, 575107, 1119210, 2031933, 3488500, .... (End)
G.f.: x*(x^12 +324*x^11 +15606*x^10 +233300*x^9 +1424925*x^8 +4050864*x^7 +5703096*x^6 +4050864*x^5 +1424925*x^4 +233300*x^3 +15606*x^2 +324*x +1) / (x -1)^20. - Colin Barker, May 02 2014
a(n) = (n/120679663104000)*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(-864000 + 2116800*n + 772737840*n^2 + 3398930472*n^3 + 6406454992 *n^4 + 6701566410*n^5 + 4302755765*n^6 + 1780394616*n^7 + 484074591*n^8 + 85975890*n^9 + 9604595*n^10 + 612612*n^11 + 17017*n^12). - G. C. Greubel, Nov 22 2017

More terms from Colin Barker, May 02 2014

A085439 a(n) = Sum_{i=1..n} binomial(i+1,2)^4.

Original entry on oeis.org

1, 82, 1378, 11378, 62003, 256484, 871140, 2550756, 6651381, 15802006, 34776742, 71791798, 140366759, 261917384, 469277384, 811379400, 1359360681, 2214396762, 3517606762, 5462416762, 8309813083, 12406965164, 18209748140, 26309748140, 37466388765, 52644875166

Offset: 1

André F. Labossière, Jul 03 2003

			a(15) = (2520*(15^9) +22680*(15^8) +79920*(15^7) +136080*(15^6) +107352*(15^5) +22680*(15^4) -10080*(15^3) +1728*15)/9! = 469277384.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 13.
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Column k=4 of A334781.

Cf. A000292, A087127, A024166, A024166, A085438, A085440, A085441, A085442, A000332, A086020, A086021, A086022, A000389, A086023, A086024, A000579, A086025, A086026, A000580, A086027, A086028, A027555, A086029, A086030.

[(2520*n^9 +22680*n^8 +79920*n^7 +136080*n^6 +107352*n^5 +22680*n^4 -10080*n^3 +1728*n)/Factorial(9): n in [1..30]]; // G. C. Greubel, Nov 22 2017

Table[(2520*(n^9) + 22680*(n^8) + 79920*(n^7) + 136080*(n^6) + 107352*(n^5) + 22680*(n^4) - 10080*(n^3) + 1728*n)/9!, {n, 1, 50}] (* G. C. Greubel, Nov 22 2017 *)

Vec(x*(x^6+72*x^5+603*x^4+1168*x^3+603*x^2+72*x+1)/(x-1)^10 + O(x^100)) \\ Colin Barker, May 02 2014

a(n) = sum(i=1, n, binomial(i+1, 2)^4); \\ Michel Marcus, Nov 22 2017

a(n) = (2520*n^9 +22680*n^8 +79920*n^7 +136080*n^6 +107352*n^5 +22680*n^4 -10080*n^3 +1728*n)/9!.

G.f.: x*(x^6+72*x^5+603*x^4+1168*x^3+603*x^2+72*x+1) / (x-1)^10. - Colin Barker, May 02 2014

More terms from Colin Barker, May 02 2014

Typo in example fixed by Colin Barker, May 02 2014

A085440 a(n) = Sum_{i=1..n} binomial(i+1,2)^5.

Original entry on oeis.org

1, 244, 8020, 108020, 867395, 4951496, 22161864, 82628040, 267156165, 770440540, 2022773116, 4909947484, 11150268935, 23913084560, 48796284560, 95322158736, 179163294729, 325374464580, 572984364580, 981394464580, 1639143014731, 2675722491224, 4277290592600

Offset: 1

André F. Labossière, Jun 30 2003

Elisabeth Busser and Gilles Cohen, Neuro-Logies - "Chercher, jouer, trouver", La Recherche, April 1999, No. 319, page 97.

T. D. Noe, Table of n, a(n) for n = 1..1000
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 13.
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Column k=5 of A334781.

Cf. A000292, A087127, A024166, A024166, A085438, A085439, A085441, A085442, A000332, A086020, A086021, A086022, A000389, A086023, A086024, A000579, A086025, A086026, A000580, A086027, A086028, A027555, A086029, A086030.

[(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 )/Factorial(11): n in [1..30]]; // G. C. Greubel, Nov 22 2017

Table[(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!, {n,1,50}] (* G. C. Greubel, Nov 22 2017 *)

for(n=1,30, print1(sum(k=1,n, binomial(k+1,2)^5), ", ")) \\ G. C. Greubel, Nov 22 2017

a(n) = (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!.

G.f.: x*(x^8+232*x^7+5158*x^6+27664*x^5+47290*x^4+27664*x^3+5158*x^2+232*x+1) / (x-1)^12. - Colin Barker, May 02 2014

Formula edited by Colin Barker, May 02 2014

A085441 a(n) = Sum_{i=1..n} binomial(i+1,2)^6.

Original entry on oeis.org

1, 730, 47386, 1047386, 12438011, 98204132, 580094436, 2756876772, 11060642397, 38741283022, 121395233038, 346594833742, 914464085783, 2254559726408, 5240543726408, 11568062614344, 24395756421273, 49397866465794, 96443747465794, 182209868465794

Offset: 1

André F. Labossière, Jul 07 2003

			a(5) = C(7,3)*[191*106 + 450*(18*C(14,10) + 3851*C(13,10) + 61839*C(12,10) + 225352*C(11,10) + 225352*C(10,10))]/10010 = 12438011.

T. D. Noe, Table of n, a(n) for n = 1..1000
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 13.
Index entries for linear recurrences with constant coefficients, signature (14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1).

Column k=6 of A334781.

Cf. A000292, A087127, A024166, A024166, A085438, A085439, A085440, A085442, A000332, A086020, A086021, A086022, A000389, A086023, A086024, A000579, A086025, A086026, A000580, A086027, A086028, A027555, A086029, A086030, A234253.

[(n/960960)*(6112 - 40040*n^2 + 78078*n^4 + 15015*n^5 + 19305*n^6 + 225225*n^7 + 335335*n^8 + 225225*n^9 + 80535*n^10 + 15015*n^11 + 1155*n^12): n in [1..30]]; // G. C. Greubel, Nov 22 2017

f:= sum(binomial(1+i,2)^6,i=1..n):
seq(f, n=1..30); # Robert Israel, Nov 22 2017

Table[Sum[Binomial[i+1,2]^6,{i,n}],{n,20}] (* or *) LinearRecurrence[ {14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1},{1,730,47386,1047386,12438011, 98204132,580094436, 2756876772,11060642397, 38741283022,121395233038, 346594833742, 914464085783, 2254559726408},20] (* Harvey P. Dale, Jun 05 2017 *)

for(n=1,30, print1(sum(k=1,n, binomial(k+1,2)^6), ", ")) \\ G. C. Greubel, Nov 22 2017

G.f.: x*(x^10 +716*x^9 +37257*x^8 +450048*x^7 +1822014*x^6 +2864328*x^5 +1822014*x^4 +450048*x^3 +37257*x^2 +716*x +1) / (x -1)^14. - Colin Barker, May 02 2014

a(n) = (n/960960)*(6112 - 40040*n^2 + 78078*n^4 + 15015*n^5 + 19305*n^6 + 225225*n^7 + 335335*n^8 + 225225*n^9 + 80535*n^10 + 15015*n^11 + 1155*n^12). - G. C. Greubel, Nov 22 2017

A086022 a(n) = Sum_{i=1..n} C(i+2,3)^4.

Original entry on oeis.org

1, 257, 10257, 170257, 1670882, 11505378, 61292514, 268652514, 1009853139, 3352413139, 10042998755, 27598188771, 70457539396, 168802499396, 382616259396, 825980472132, 1707628231653, 3396588391653, 6525595601653, 12150082161653, 21987344308134, 38769279231910

Offset: 1

André F. Labossière, Jul 11 2003

			a(8) = C(11,4)*[-41*2793 + 350*(47*C(16,9) + 1749*C(15,9) + 9292*C(14,9) + 9292*C(13,9) + 1749*C(12,9) + 47*C(11,9))]/15015 = 268652514 .

T. D. Noe, Table of n, a(n) for n = 1..1000
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 13.
Index entries for linear recurrences with constant coefficients, signature (14, -91, 364, -1001, 2002, -3003, 3432, -3003, 2002, -1001, 364, -91, 14, -1).

Cf. A086020, A086021, A086023, A086024, A086025, A086026, A086027, A086028, A086029, A086030, A087127, A024166, A085438, A085439, A085440, A085441, A085442.

[(n/12972960)*(-8856 +60060*n^2 +165165*n^3 +841841*n^4 +2462460*n^5 +3709420*n^6 +3243240*n^7 +1756755*n^8 +600600*n^9 +126490*n^10 +15015*n^11 +770*n^12): n in [1..30]]; // G. C. Greubel, Nov 22 2017

Accumulate[Binomial[Range[3,30],3]^4] (* Harvey P. Dale, Oct 09 2016 *)

for(n=1,30, print1((n/12972960)*(-8856 + 60060*n^2 + 165165*n^3 + 841841*n^4 + 2462460*n^5 + 3709420*n^6 + 3243240*n^7 + 1756755*n^8 + 600600*n^9 + 126490*n^10 + 15015*n^11 + 770*n^12), ", ")) \\ G. C. Greubel, Nov 22 2017

G.f.: x*(1+x)*(x^8 +242*x^7 +6508*x^6 +43174*x^5 +84950*x^4 +43174*x^3 +6508*x^2 +242*x + 1) / (x-1)^14 . - R. J. Mathar, Dec 22 2013
(n-1)^4*a(n) +(-2*n^4 -4*n^3 -30*n^2 -28*n -17)*a(n-1) +(n+2)^4*a(n-2)=0. - R. J. Mathar, Dec 22 2013
a(n) = C(n+3,4)*[-41*F3(n) +350*(47*C(n+8,9) + 1749*C(n+7,9) + 9292*C(n+6,9) + 9292*C(n+5,9) + 1749*C(n+4,9) + 47*C(n+3,9))]/15015, where F3(n) = -C(3,0)*C(n+3,0) + C(4,1)*C(n+3,1) - C(5,2)*C(n+3,2) + C(6,3)*C(n+3,3). The value of F3(n), (n=0..8) is: 1, 35, 119, 273, 517, 871, 1355, 1989, 2793, ... - Yahia Kahloune, Dec 23 2013
a(n) = (n/12972960)*(-8856 + 60060*n^2 + 165165*n^3 + 841841*n^4 + 2462460*n^5 + 3709420*n^6 + 3243240*n^7 + 1756755*n^8 + 600600*n^9 + 126490*n^10 + 15015*n^11 + 770*n^12). - G. C. Greubel, Nov 22 2017

cp's OEIS Frontend

A086024 a(n) = Sum_{i=1..n} C(i+3,4)^3.

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A086029 a(n) = Sum_{i=1..n} C(i+6,7)^2.

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A086021 a(n) = Sum_{i=1..n} C(i+2,3)^3.

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A086026 a(n) = Sum_{i=1..n} C(i+4,5)^3.

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A086027 a(n) = Sum_{i=1..n} binomial(i+5,6)^2.

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A086028 a(n) = Sum_{i=1..n} C(i+5,6)^3.

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