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A309010
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Square array A(n, k) = Sum_{j=0..n} binomial(n,j)^k, n >= 0, k >= 0, read by antidiagonals.
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12
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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
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OFFSET
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0,3
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COMMENTS
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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
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REFERENCES
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R. P. Stanley, Enumerative Combinatorics Vol I, Second Edition, Cambridge, 2011, Example 3.18.3 d, page 366.
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LINKS
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FORMULA
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A(n, k) = Sum_{j=0..n} binomial(n,j)^k (array).
T(n, k) = A(k, n-k) (antidiagonals).
Sum_{n>=0} A(n,k) x^n/(n!^k) = (Sum_{n>=0} x^n/(n!^k))^2. - Geoffrey Critzer, Apr 17 2020
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EXAMPLE
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Square array, A(n, k), begins:
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;
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Columns k=0..12 give A000027(n+1), A000079, A000984, A000172, A005260, A005261, A069865, A182421, A182422, A182446, A182447, A342294, A342295.
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KEYWORD
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AUTHOR
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STATUS
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approved
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