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

0,3

Comments

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

References

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

Links

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

Formula

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

Example

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;

Mathematica

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 *)

Prog

(PARI) A(n, k) = sum(j=0, n, binomial(n, j)^k); \\ Seiichi Manyama, Jan 08 2022
(Magma) [(&+[Binomial(k, j)^(n-k): j in [0..k]]): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 26 2022
(SageMath) 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

Crossrefs

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.

Sequence in context: A104795 A347570 A116925 * A308500 A210950 A214314

Adjacent sequences: A309007 A309008 A309009 * A309011 A309012 A309013

Keyword

nonn,tabl

Author

Seiichi Manyama, Jul 06 2019

Status

approved