A245397 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.
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
Offset: 0
Examples
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, ...
Links
- Alois P. Heinz, Antidiagonals n = 0..50, flattened
Crossrefs
Programs
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Maple
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); -
Mathematica
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 *)
Formula
A(n,k) = [x^n] (n!)^k * (Sum_{j=0..n} x^j/(j!)^k)^n.