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A245397
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A(n,k) is the sum of k-th powers of coefficients in full expansion of (z_1+z_2+...+z_n)^n; square array A(n,k), n>=0, k>=0, read by antidiagonals.
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12
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1, 1, 1, 1, 1, 3, 1, 1, 4, 10, 1, 1, 6, 27, 35, 1, 1, 10, 93, 256, 126, 1, 1, 18, 381, 2716, 3125, 462, 1, 1, 34, 1785, 36628, 127905, 46656, 1716, 1, 1, 66, 9237, 591460, 7120505, 8848236, 823543, 6435, 1, 1, 130, 51033, 11007556, 495872505, 2443835736, 844691407, 16777216, 24310
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OFFSET
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0,6
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LINKS
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FORMULA
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A(n,k) = [x^n] (n!)^k * (Sum_{j=0..n} x^j/(j!)^k)^n.
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EXAMPLE
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A(3,2) = 93: (z1+z2+z3)^3 = z1^3 +3*z1^2*z2 +3*z1^2*z3 +3*z1*z2^2 +6*z1*z2*z3 +3*z1*z3^2 +z2^3 +3*z2^2*z3 +3*z2*z3^2 +z3^3 => 1^2+3^2+3^2+3^2+6^2+3^2+1^2+3^2+3^2+1^2 = 93.
Square array A(n,k) begins:
0 : 1, 1, 1, 1, 1, 1, ...
1 : 1, 1, 1, 1, 1, 1, ...
2 : 3, 4, 6, 10, 18, 34, ...
3 : 10, 27, 93, 381, 1785, 9237, ...
4 : 35, 256, 2716, 36628, 591460, 11007556, ...
5 : 126, 3125, 127905, 7120505, 495872505, 41262262505, ...
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MAPLE
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b:= proc(n, i, k) option remember; `if`(n=0 or i=1, 1,
add(b(n-j, i-1, k)*binomial(n, j)^k, j=0..n))
end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..10);
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MATHEMATICA
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b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, Sum[b[n-j, i-1, k] * Binomial[n, j]^(k-1)/j!, {j, 0, n}]]]; A[n_, k_] := n!*b[n, n, k]; Table[ Table[A[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Jan 30 2015, after Alois P. Heinz *)
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CROSSREFS
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Columns k=0-10 give: A001700(n-1) for n>0, A000312, A033935, A055733, A055740, A246240, A246241, A246242, A246243, A246244, A246245.
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KEYWORD
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AUTHOR
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STATUS
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approved
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