A069865 -id:A069865 - 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

A382842 a(n) = Sum_{k=0..floor(n/2)} (binomial(n,k) * binomial(n-k,k))^3.

Original entry on oeis.org

1, 1, 9, 217, 1945, 35001, 764001, 12079089, 250222617, 5424133465, 107360983009, 2358751625649, 52540471866961, 1147794435985393, 26151265459123065, 600227875293254217, 13779170435209475097, 322302377797126709913, 7582484532013652243169, 179184911648568670363185, 4275721755296040840336945

Offset: 0

Ilya Gutkovskiy, Apr 06 2025

nonn

Diagonal of the rational function 1 / ((1 - x)*(1 - y)*(1 - z)*(1 - u)*(1 - v)*(1 - w) - (x*y*z)^2*u*v*w).

Cf. A000172, A002426, A069865, A092813, A181545, A382841.

Cf. A089627.

a:= n-> add(combinat[multinomial](n, n-2*k, k$2)^3, k=0..n/2):
seq(a(n), n=0..20);  # Alois P. Heinz, Apr 07 2025

Table[Sum[(Binomial[n, k] Binomial[n - k, k])^3, {k, 0, Floor[n/2]}], {n, 0, 20}]
Table[HypergeometricPFQ[{1/2 - n/2, 1/2 - n/2, 1/2 - n/2, -n/2, -n/2, -n/2}, {1, 1, 1, 1, 1}, 64], {n, 0, 20}]
Table[SeriesCoefficient[1/((1 - x) (1 - y) (1 - z) (1 - u) (1 - v) (1 - w) - (x y z)^2 u v w), {x, 0, n}, {y, 0, n}, {z, 0, n}, {u, 0, n}, {v, 0, n}, {w, 0, n}], {n, 0, 20}]

a(n) ~ 3^(3*n+3) / (8 * Pi^(5/2) * n^(5/2)). - Vaclav Kotesovec, Apr 07 2025

a(n) = Sum_{k=0..floor(n/2)} A089627(n,k)^3. - Alois P. Heinz, Apr 07 2025

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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A382842 a(n) = Sum_{k=0..floor(n/2)} (binomial(n,k) * binomial(n-k,k))^3.

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