A005261 -id:A005261 - cp's OEIS frontend

A309010 Square array A(n, k) = Sum_{j=0..n} binomial(n,j)^k, n >= 0, k >= 0, read by antidiagonals.

1, 1, 2, 1, 2, 3, 1, 2, 4, 4, 1, 2, 6, 8, 5, 1, 2, 10, 20, 16, 6, 1, 2, 18, 56, 70, 32, 7, 1, 2, 34, 164, 346, 252, 64, 8, 1, 2, 66, 488, 1810, 2252, 924, 128, 9, 1, 2, 130, 1460, 9826, 21252, 15184, 3432, 256, 10, 1, 2, 258, 4376, 54850, 206252, 263844, 104960, 12870, 512, 11

Offset: 0

Seiichi Manyama, Jul 06 2019

A(n,k) is the constant term in the expansion of (Product_{j=1..k-1} (1 + x_j) + Product_{j=1..k-1} (1 + 1/x_j))^n for k > 0. - Seiichi Manyama, Oct 27 2019
Let B_k be the binomial poset containing all k-tuples of equinumerous subsets of {1,2,...} ordered by inclusion componentwise (described in Stanley reference below).  Then A(k,n) is the number of elements in any n-interval of B_k. - Geoffrey Critzer, Apr 16 2020
Column k is the diagonal of the rational function 1 / (Product_{j=1..k} (1-x_j) - Product_{j=1..k} x_j) for k>0. - Seiichi Manyama, Jul 11 2020

			Square array, A(n, k), begins:
   1,  1,   1,    1,     1,      1, ... A000012;
   2,  2,   2,    2,     2,      2, ... A007395;
   3,  4,   6,   10,    18,     34, ... A052548;
   4,  8,  20,   56,   164,    488, ... A115099;
   5, 16,  70,  346,  1810,   9826, ...
   6, 32, 252, 2252, 21252, 206252, ...
Antidiagonals, T(n, k), begin:
  1;
  1,  2;
  1,  2,   3;
  1,  2,   4,    4;
  1,  2,   6,    8,    5;
  1,  2,  10,   20,   16,     6;
  1,  2,  18,   56,   70,    32,     7;
  1,  2,  34,  164,  346,   252,    64,    8;
  1,  2,  66,  488, 1810,  2252,   924,  128,   9;
  1,  2, 130, 1460, 9826, 21252, 15184, 3432, 256,  10;

R. P. Stanley, Enumerative Combinatorics Vol I, Second Edition, Cambridge, 2011, Example 3.18.3 d, page 366.

Seiichi Manyama, Antidiagonals n = 0..100, flattened

Columns k=0..12 give A000027(n+1), A000079, A000984, A000172, A005260, A005261, A069865, A182421, A182422, A182446, A182447, A342294, A342295.
Main diagonal gives A167010.
Cf. A328747, A328748, A328807, A328812.
Cf. A000012, A007395, A052548, A115099.

[(&+[Binomial(k,j)^(n-k): j in [0..k]]): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 26 2022

nn = 8; Table[ek[x_] := Sum[x^n/n!^k, {n, 0, nn}];Range[0, nn]!^k CoefficientList[Series[ek[x]^2, {x, 0, nn}],x], {k, 0, nn}] // Transpose // Grid (* Geoffrey Critzer, Apr 17 2020 *)

A(n, k) = sum(j=0, n, binomial(n, j)^k); \\ Seiichi Manyama, Jan 08 2022

flatten([[sum(binomial(k,j)^(n-k) for j in (0..k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Aug 26 2022

A(n, k) = Sum_{j=0..n} binomial(n,j)^k (array).
A(n, n+1) = A328812(n).
A(n, n) = A167010(n).
T(n, k) = A(k, n-k) (antidiagonals).
T(n, n) = A000027(n+1).
T(n, n-1) = A000079(n-1).
T(n, n-2) = A000984(n-2).
T(n, n-3) = A000172(n-3).
T(n, n-4) = A005260(n-4).
T(n, n-5) = A005261(n-5).
T(n, n-6) = A069865(n-6).
T(n, n-7) = A182421(n-7).
T(n, n-8) = A182422(n-8).
T(n, n-9) = A182446(n-9).
T(n, n-10) = A182447(n-10).
T(n, n-11) = A342294(n-11).
T(n, n-12) = A342295(n-12).
Sum_{n>=0} A(n,k) x^n/(n!^k) = (Sum_{n>=0} x^n/(n!^k))^2. - Geoffrey Critzer, Apr 17 2020

A342294 a(n) = Sum_{k = 0..n} binomial(n,k)^11.

Original entry on oeis.org

1, 2, 2050, 354296, 371185666, 200097656252, 222100237312864, 193798873701831680, 231719476114879600642, 257097895846251291074612, 330463219813679264204224300, 419460465362069257397304825200, 573863850341313751827291703127200

Offset: 0

N. J. A. Sloane, Mar 27 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..306
M. A. Perlstadt, Some Recurrences for Sums of Powers of Binomial Coefficients, Journal of Number Theory 27 (1987), pp. 304-309.

Column 11 of A309010.

Sum_{k = 0..n} C(n,k)^m for m = 1..12: A000079, A000984, A000172, A005260, A005261, A069865, A182421, A182422, A182446, A182447, A342294, A342295.

Table[Sum[Binomial[n,k]^11,{k,0,n}],{n,0,15}] (* Harvey P. Dale, May 08 2025 *)

a(n) = sum(k=0, n, binomial(n, k)^11); \\ Michel Marcus, Mar 27 2021

a(n) ~ 2^(p*n)/sqrt(p) * (2/(Pi*n))^((p-1)/2) * (1 - (p-1)^2/(4*p*n)), set p=11. - Vaclav Kotesovec, Aug 04 2022

A342295 a(n) = Sum_{k = 0..n} binomial(n,k)^12.

Original entry on oeis.org

1, 2, 4098, 1062884, 2210336770, 2000488281252, 4355497029345924, 6773152698818628936, 15744083665278445490178, 32270900877696351763796420, 80314784333143089874093429348, 192454957455454582636391397662856, 509571049488109525160616367158261124

Offset: 0

N. J. A. Sloane, Mar 27 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..281
M. A. Perlstadt, Some Recurrences for Sums of Powers of Binomial Coefficients, Journal of Number Theory 27 (1987), pp. 304-309.

Column 12 of A309010.

Sum_{k = 0..n} C(n,k)^m for m = 1..12: A000079, A000984, A000172, A005260, A005261, A069865, A182421, A182422, A182446, A182447, A342294, A342295.

a(n) = sum(k=0, n, binomial(n, k)^12); \\ Michel Marcus, Mar 27 2021

a(n) ~ 2^(p*n)/sqrt(p) * (2/(Pi*n))^((p-1)/2) * (1 - (p-1)^2/(4*p*n)), set p=12. - Vaclav Kotesovec, Aug 04 2022

A328750 Constant term in the expansion of (-1 + (1 + w) * (1 + x) * (1 + y) * (1 + z) + (1 + 1/w) * (1 + 1/x) * (1 + 1/y) * (1 + 1/z))^n.

Original entry on oeis.org

1, 1, 31, 391, 8071, 161671, 3634921, 84109201, 2032357111, 50355327991, 1277302604521, 32983865502721, 864982811998801, 22976755021842961, 617140285389771391, 16735405610179740151, 457647302453165769751, 12607719926638032161431, 349620344754345216824041

Offset: 0

Seiichi Manyama, Oct 27 2019

nonn

Column k=5 of A328747.

Cf. A005261, A328751.

Table[Sum[(-1)^(n - i)*Binomial[n, i]*Sum[Binomial[i, j]^5, {j, 0, i}], {i, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 28 2019 *)

{a(n) = sum(i=0, n, (-1)^(n-i)*binomial(n,i)*sum(j=0, i, binomial(i, j)^5))}

a(n) = Sum_{i=0..n} (-1)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^5.
From Vaclav Kotesovec, Oct 28 2019: (Start)
Recurrence: n^4*(440*n^2 - 2728*n + 3723)*a(n) = (6600*n^6 - 54120*n^5 + 147013*n^4 - 174348*n^3 + 102442*n^2 - 29260*n + 3108)*a(n-1) + (194920*n^6 - 1988184*n^5 + 7650713*n^4 - 14588908*n^3 + 14793198*n^2 - 7658420*n + 1601964)*a(n-2) + (n-2)*(690800*n^5 - 7046160*n^4 + 26712814*n^3 - 47822370*n^2 + 40779795*n - 13361628)*a(n-3) + (n-3)*(n-2)*(975480*n^4 - 8974416*n^3 + 28602923*n^2 - 37477643*n + 16905924)*a(n-4) + (n-4)*(n-3)*(n-2)*(622600*n^3 - 4482720*n^2 + 9455173*n - 5628497)*a(n-5) + 341*(n-5)*(n-4)*(n-3)*(n-2)*(440*n^2 - 1848*n + 1435)*a(n-6).
a(n) ~ 31^(n+2) / (256 * sqrt(5) * Pi^2 * n^2). (End)

A328751 Constant term in the expansion of (-2 + (1 + w) * (1 + x) * (1 + y) * (1 + z) + (1 + 1/w) * (1 + 1/x) * (1 + 1/y) * (1 + 1/z))^n.

Original entry on oeis.org

1, 0, 30, 300, 6690, 124920, 2778600, 61790400, 1452751650, 34806097200, 855836532180, 21393889763400, 543342862524000, 13972938142363200, 363356617578926400, 9538720137580233600, 252510537115100657250, 6733792260826534332000, 180751978201192700659500

Offset: 0

Seiichi Manyama, Oct 27 2019

nonn

Column k=5 of A328748.

Cf. A005261, A328750.

Table[Sum[(-2)^(n-i)*Binomial[n,i] * Sum[Binomial[i,j]^5, {j,0,i}], {i,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 20 2023 *)

{a(n) = sum(i=0, n, (-2)^(n-i)*binomial(n, i)*sum(j=0, i, binomial(i, j)^5))}

a(n) = Sum_{i=0..n} (-2)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^5.
From Vaclav Kotesovec, Mar 20 2023: (Start)
Recurrence: n^4*(22*n^2 - 198*n + 323)*a(n) = (n-1)*(198*n^5 - 1980*n^4 + 4535*n^3 - 2641*n^2 + 119*n + 210)*a(n-1) + (11066*n^6 - 143858*n^5 + 628715*n^4 - 1298438*n^3 + 1394723*n^2 - 756728*n + 165060)*a(n-2) + 4*(n-2)*(19096*n^5 - 248248*n^4 + 1086158*n^3 - 2156993*n^2 + 2004912*n - 708435)*a(n-3) + 40*(n-3)*(n-2)*(5346*n^4 - 64152*n^3 + 242653*n^2 - 363566*n + 182959)*a(n-4) + 400*(n-4)*(n-3)*(n-2)*(682*n^3 - 6820*n^2 + 17955*n - 12432)*a(n-5) + 6000*(n-5)*(n-4)*(n-3)*(n-2)*(22*n^2 - 154*n + 147)*a(n-6).
a(n) ~ 2^(n-6) * 3^(n+2) * 5^(n + 3/2) / (Pi^2 * n^2). (End)

A180350 G.f.: Sum_{n>=0} a(n)*x^n/n!^5 = [ Sum_{n>=0} x^n/n!^5 ]^3.

Original entry on oeis.org

1, 3, 99, 9237, 775971, 83118753, 10657602909, 1463886204147, 215566192274211, 33677584957306713, 5492032622227428849, 928229455634614797447, 161727023896151286167901, 28905146810167510775300463

Offset: 0

Paul D. Hanna, Jan 20 2011

nonn

			G.f.: A(x) = 1 + 3*x + 99*x^2/2!^5 + 9237*x^3/3!^5 + 775971*x^4/4!^5 +...
A(x)^(1/3) = 1 + x + x^2/2!^5 + x^3/3!^5 + x^4/4!^5 +...

Seiichi Manyama, Table of n, a(n) for n = 0..424

Cf. A002893, A005261, A141057, A172434, A336270.

{a(n)=if(n<0, 0, n!^5*polcoeff(sum(m=0, n, x^m/m!^5+x*O(x^n))^3, n))}

{a(n)=sum(k=0, n, binomial(n, k)^5*sum(j=0, k, binomial(k, j)^5))}

a(n) = Sum_{k=0..n} C(n,k)^5 * Sum_{j=0..k} C(k,j)^5 = Sum_{k=0..n} C(n,k)^5 * A005261(k).

A374614 a(n) = Sum_{k=0..n} (k/n)^2 * binomial(n,k)^5.

Original entry on oeis.org

1, 9, 136, 2585, 54126, 1227492, 29226688, 723533337, 18438032890, 480994824134, 12787403151744, 345355150592036, 9451729196625184, 261628075707534720, 7313361005558843136, 206190939973811373593, 5857313490484652859282

Offset: 1

Seiichi Manyama, Jul 14 2024

nonn

Cf. A005261, A374615, A374616.

Cf. A181067, A198256.

Table[Sum[(k/n)^2 Binomial[n,k]^5,{k,0,n}],{n,20}] (* Harvey P. Dale, Jun 16 2025 *)

a(n) = sum(k=0, n-1, binomial(n-1, k)^2*binomial(n, k)^3);

a(n) = Sum_{k=0..n-1} binomial(n-1,k)^2 * binomial(n,k)^3.

A374615 a(n) = Sum_{k=0..n} (k/n)^3 * binomial(n,k)^5.

Original entry on oeis.org

1, 5, 82, 1421, 29626, 657662, 15528640, 381137549, 9656742322, 250689517130, 6638957500924, 178721359853390, 4878005765458528, 134712060315562784, 3758101325718600832, 105769714118196065933, 3000003700599260555650, 85677293959381174518986

Offset: 1

Seiichi Manyama, Jul 14 2024

nonn

Cf. A005261, A374614, A374616.

Cf. A181069.

a(n) = sum(k=0, n-1, binomial(n-1, k)^3*binomial(n, k)^2);

a(n) = Sum_{k=0..n-1} binomial(n-1,k)^3 * binomial(n,k)^2.

A374616 a(n) = Sum_{k=0..n} (k/n)^4 * binomial(n,k)^5.

Original entry on oeis.org

1, 3, 52, 815, 16806, 363132, 8471296, 205501599, 5164447210, 133153140098, 3506969720904, 93973493327012, 2554956958697248, 70323373958110080, 1956176944656294912, 54916687591986040223, 1554166975730511463794, 44297812047491490990366

Offset: 1

Seiichi Manyama, Jul 14 2024

nonn

Cf. A005261, A374614, A374615.

a(n) = sum(k=0, n-1, binomial(n-1, k)^4*binomial(n, k));

a(n) = Sum_{k=0..n-1} binomial(n-1,k)^4 * binomial(n,k).

A328809 Constant term in the expansion of (1 + (1 + w) * (1 + x) * (1 + y) * (1 + z) + (1 + 1/w) * (1 + 1/x) * (1 + 1/y) * (1 + 1/z))^n.

Original entry on oeis.org

1, 3, 39, 597, 11991, 260613, 6129489, 151078707, 3867441111, 101852866533, 2744610170049, 75348380209347, 2100889194001761, 59349600029522403, 1695505948476461559, 48909452234258070117, 1422877722974198091351, 41704912707174877940613

Offset: 0

Seiichi Manyama, Oct 28 2019

nonn

Column k=5 of A328807.

Cf. A005261, A328750, A328751.

Table[Sum[Binomial[n, i]*Sum[Binomial[i, j]^5, {j, 0, i}], {i, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 28 2019 *)

{a(n) = sum(i=0, n, binomial(n, i)*sum(j=0, i, binomial(i, j)^5))}

a(n) = Sum_{i=0..n} binomial(n,i)*Sum_{j=0..i} binomial(i,j)^5.
From Vaclav Kotesovec, Oct 28 2019: (Start)
Recurrence: n^4*(40*n^2 - 24*n - 79)*a(n) = (1080*n^6 - 2808*n^5 + 875*n^4 + 2928*n^3 - 3762*n^2 + 1834*n - 336)*a(n-1) + (9320*n^6 - 42872*n^5 + 61193*n^4 - 12152*n^3 - 35518*n^2 + 21658*n - 2016)*a(n-2) - (n-2)*(48560*n^5 - 223376*n^4 + 216118*n^3 + 381866*n^2 - 791133*n + 355194)*a(n-3) + (n-3)*(n-2)*(79560*n^4 - 286416*n^3 - 56675*n^2 + 976675*n - 616322)*a(n-4) - 11*(n-4)*(n-3)*(n-2)*(5080*n^3 - 8128*n^2 - 25641*n + 21693)*a(n-5) + 363*(n-5)*(n-4)*(n-3)*(n-2)*(40*n^2 + 56*n - 63)*a(n-6).
a(n) ~ 33^(n+2) / (256 * sqrt(5) * Pi^2 * n^2). (End)

cp's OEIS Frontend

A309010 Square array A(n, k) = Sum_{j=0..n} binomial(n,j)^k, n >= 0, k >= 0, read by antidiagonals.

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A342294 a(n) = Sum_{k = 0..n} binomial(n,k)^11.

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A342295 a(n) = Sum_{k = 0..n} binomial(n,k)^12.

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A328750 Constant term in the expansion of (-1 + (1 + w) * (1 + x) * (1 + y) * (1 + z) + (1 + 1/w) * (1 + 1/x) * (1 + 1/y) * (1 + 1/z))^n.

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A328751 Constant term in the expansion of (-2 + (1 + w) * (1 + x) * (1 + y) * (1 + z) + (1 + 1/w) * (1 + 1/x) * (1 + 1/y) * (1 + 1/z))^n.

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A180350 G.f.: Sum_{n>=0} a(n)*x^n/n!^5 = [ Sum_{n>=0} x^n/n!^5 ]^3.

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A374614 a(n) = Sum_{k=0..n} (k/n)^2 * binomial(n,k)^5.

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A374615 a(n) = Sum_{k=0..n} (k/n)^3 * binomial(n,k)^5.

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