A086020 -id:A086020 - cp's OEIS frontend

A001249 Squares of tetrahedral numbers: a(n) = binomial(n+3,n)^2.

1, 16, 100, 400, 1225, 3136, 7056, 14400, 27225, 48400, 81796, 132496, 207025, 313600, 462400, 665856, 938961, 1299600, 1768900, 2371600, 3136441, 4096576, 5290000, 6760000, 8555625, 10732176, 13351716, 16483600, 20205025, 24601600, 29767936, 35808256

Offset: 0

N. J. A. Sloane

Total area of all square and rectangular regions from an n+1 X n+1 grid. E.g., n = 2, there are 9 individual squares, 4 2 X 2's and 1 3 X 3, total area 9 + 16 + 9 = 34. The rectangular regions include 6 2 X 1's, 6 1 X 2's, 3 3 X 1's, 3 1 X 3's, 2 3 X 2's and 2 2 X 3's, total area 12 + 12 + 9 + 9 + 12 + 12 = 66, hence a(2) = 34 + 66 = 100. - Jon Perry, Jul 29 2003 [Index/grid size adjusted by Rick L. Shepherd, Jun 27 2017]
Number of 3 X 3 submatrices of an n+3 X n+3 matrix. - Rick L. Shepherd, Jun 27 2017
The inverse binomial transform gives row n=2 of A087107. - R. J. Mathar, Aug 31 2022

T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000290, A000292, A006542, A033455, A108674 (first diffs.), A086020 (partial sums).
Third column of triangle A008459.
Cf. A000579, A001303, A040977.

A001249 := proc(n) binomial(n+3,n)^2 end proc: seq(A001249(n),n=0..10) ; # Zerinvary Lajos, May 17 2006

Table[Binomial[n + 3, 3]^2, {n, 0, 100}] (* T. D. Noe, Jun 26 2012 *)

a(n)=binomial(n+3,3)^2 \\ Charles R Greathouse IV, Sep 24 2015

From R. J. Mathar, Aug 19 2008: (Start)
a(n) = (A000292(n+1))^2.
O.g.f.: (1+x)(x^2+8x+1)/(1-x)^7. (End)
a(n) = C(n+4, 3)*C(n+4, 4)/(n+4) + A001303(n) = C(n+4, 3)*C(n+3, 3)/4 + A001303(n) = C(n+4, 6) + 3*C(n+5, 6) + C(n+6,6) + A001303(n). - Gary Detlefs, Aug 07 2013
-n^2*a(n) + (n+3)^2*a(n-1) = 0. - R. J. Mathar, Aug 15 2013
a(n) = 9*A040977(n-1) + A000579(n+6) + A000579(n+3). - R. J. Mathar, Aug 15 2013
a(n) = (n+3)*C(n+2, 2)*C(n+3, 3)/3. - Gary Detlefs, Jan 06 2014
a(n) = A000290(n+1)*A000290(n+2)*A000290(n+3)/36. - Bruno Berselli, Nov 12 2014
G.f. 2F1(4,4;1;x). - R. J. Mathar, Aug 09 2015
E.g.f.: exp(x)*(1 + 15*x + 69*x^2/2! + 147*x^3/3! + 162*x^4/4! +  90*x^5/5! +  20*x^6/6!). Computed from the o.g.f with the formulas (23) - (25) of the W. Lang link given in A060187. - Wolfdieter Lang, Jul 27 2017
From Amiram Eldar, Jan 24 2022: (Start)
Sum_{n>=0} 1/a(n) = 9*Pi^2 - 351/4.
Sum_{n>=0} (-1)^n/a(n) = 63/4 - 3*Pi^2/2. (End)
a(n) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7). - Wesley Ivan Hurt, Aug 29 2022
a(n) = a(n-1)+A000217(n+1)*A000330(n+1). - J. M. Bergot, Aug 29 2022
a(n) = A002415(n+2)^2 - 20*A006857(n-1). - Yasser Arath Chavez Reyes, Nov 08 2024

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

A174266 Irregular triangle read by rows: T(n,k) = Sum_{i=0..k} (-1)^i * binomial(3n+1,i) binomial(k+3-i,3)^n, 0 <= k <= 3*(n-1).

Original entry on oeis.org

1, 1, 9, 9, 1, 1, 54, 405, 760, 405, 54, 1, 1, 243, 6750, 49682, 128124, 128124, 49682, 6750, 243, 1, 1, 1008, 83736, 1722320, 12750255, 40241088, 58571184, 40241088, 12750255, 1722320, 83736, 1008, 1, 1, 4077, 922347, 45699447, 789300477, 5904797049, 21475242671, 40396577931, 40396577931, 21475242671, 5904797049, 789300477, 45699447, 922347, 4077, 1

Offset: 1

Roger L. Bagula, Mar 14 2010

From Yahia Kahloune, Jan 30 2014: (Start)
In general, let b(k,e,p) = Sum_{i=0..k} (-1)^i*binomial(e*p+1,i)*binomial(k+e-i,e)^p. Then T(n,k) = b(k,3,n).
With these coefficients we can calculate: Sum_{i=1..n} binomial(i+e-1,e)^p = Sum_{i=0..e*(p-1)} b(i,e,p)*binomial(n+e+i,e*p+1).
For example, A086020(n) = Sum_{i=1..n} binomial(2+i, 3)^2 = T(2,0)*binomial(n+3, 7) + T(2,1)*binomial(n+4,7) + T(2,2)*binomial(n+5,7) + T(2,3)*binomial(n+6,7) = (1/5040)*(20*n^7 + 210*n^6 + 854*n^5 + 1680*n^4 + 1610*n^3 + 630*n^2 + 36*n). (End)
T(n,k) is the number of permutations of 3 indistinguishable copies of 1..n with exactly k descents. A descent is a pair of adjacent elements with the second element less than the first. - Andrew Howroyd, May 06 2020

			Triangle begins:
  1;
  1,      9,         9,            1;
  1,     54,       405,          760,            405,       54,        1;
  1,    243,      6750,        49682,         128124,   128124,    49682, ... ;
  1,   1008,     83736,      1722320,       12750255, 40241088, 58571184, ... ;
  1,   4077,    922347,     45699447,      789300477, ... ;
  1,  16362,   9639783,   1063783164,    38464072830, ... ;
  1,  65511,  98361900,  23119658500,  1641724670475, ... ;
  1, 262116, 992660346, 484099087156, 64856779908606, ... ;
...
The T(2,1) = 9 permutations of 111222 with 1 descent are: 112221, 112212, 112122, 122211, 122112, 121122, 222111, 221112, 211122. - _Andrew Howroyd_, May 07 2020

Andrew Howroyd, Table of n, a(n) for n = 1..1335
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. 12.

Columns k=0..9 are A000012, A289254, A151632, A151633, A151634, A151635, A151636, A151637, A151638, A151639.
Row sums are A014606.
Similar triangles for e=1..6: A173018 (or A008292), A154283, this sequence, A236463, A237202, A237252.
Cf. A060187, A174264.

(* First program *)
p[n_, x_]:= p[n,x]= (1-x)^(3*n+1)*Sum[(Binomial[k+1, 3])^n*x^k, {k, 0, Infinity}]/x^2;
Table[CoefficientList[p[x, n], x], {n,10}]//Flatten (* corrected by G. C. Greubel, Mar 26 2022 *)
(* Second program *)
T[n_, k_]:= T[n, k]= Sum[(-1)^(k-j+1)*Binomial[3*n+1, k-j+1]*(j*(j^2-1)/2)^n, {j, 0, k+1}]/(3^n);
Table[T[n, k], {n,10}, {k,3*n-2}]//Flatten (* G. C. Greubel, Mar 26 2022 *)

T(n,k)={sum(i=0, k, (-1)^i*binomial(3*n+1, i)*binomial(k+3-i, 3)^n)} \\ Andrew Howroyd, May 06 2020

@CachedFunction
def T(n, k): return (1/3^n)*sum( (-1)^(k-j+1)*binomial(3*n+1, k-j+1)*(j*(j^2-1)/2)^n for j in (0..k+1) )
flatten([[T(n, k) for k in (1..3*n-2)] for n in (1..10)]) # G. C. Greubel, Mar 26 2022

T(n,k) = [x^k] (1-x)^(3*n+1)*(Sum_{k>=0} (k*(k+1)*(k-1)/2)^n*x^k)/(3^n*x^2).
T(n,k) = T(n, 3*n-k).
From Yahia Kahloune, Jan 30 2014: (Start)
Sum_{i=1..n} binomial(2+i,3)^p = Sum_{i=0..3*p-3} T(p,i)*binomial(n+3+i,3*p+1).
binomial(n,3)^p = Sum_{i=0..3*p-3} T(p,i)*binomial(n+i,3*p). (End)
From Sergii Voloshyn, Dec 18 2024: (Start)
Let E be the operator (x^2)D*(1/x)*D*(x^2)*D, where D denotes the derivative operator d/dx. Then (1/6^n)* E^n(x^2/(1 - x)^4) = (row n generating polynomial)/(1 - x)^(3*n+4) = Sum_{i=0..k} (-1)^i * binomial(3*n+1,i) * binomial(k+3-i,3)^n.
For example, when n = 3 we have 1/216*E^3(x^2/(1 - x)^4) = x^2 (1 + 243x + 6750x^2 + 49682x^3 + 128124x^4 + 128124x^5 + 49682x^6 + 6750x^7 + 243x^8 + x^9)/(1 - x)^13. (End)

Edited by Andrew Howroyd, May 06 2020

cp's OEIS Frontend

A001249 Squares of tetrahedral numbers: a(n) = binomial(n+3,n)^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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A085439 a(n) = Sum_{i=1..n} binomial(i+1,2)^4.

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A174266 Irregular triangle read by rows: T(n,k) = Sum_{i=0..k} (-1)^i * binomial(3n+1,i) binomial(k+3-i,3)^n, 0 <= k <= 3*(n-1).