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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.

%I A245397 #19 Jan 27 2023 20:49:56
%S A245397 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,
%T A245397 3125,462,1,1,34,1785,36628,127905,46656,1716,1,1,66,9237,591460,
%U A245397 7120505,8848236,823543,6435,1,1,130,51033,11007556,495872505,2443835736,844691407,16777216,24310
%N 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.
%H A245397 Alois P. Heinz, <a href="/A245397/b245397.txt">Antidiagonals n = 0..50, flattened</a>
%F A245397 A(n,k) = [x^n] (n!)^k * (Sum_{j=0..n} x^j/(j!)^k)^n.
%e A245397 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.
%e A245397 Square array A(n,k) begins:
%e A245397 0 :    1,    1,      1,       1,         1,           1, ...
%e A245397 1 :    1,    1,      1,       1,         1,           1, ...
%e A245397 2 :    3,    4,      6,      10,        18,          34, ...
%e A245397 3 :   10,   27,     93,     381,      1785,        9237, ...
%e A245397 4 :   35,  256,   2716,   36628,    591460,    11007556, ...
%e A245397 5 :  126, 3125, 127905, 7120505, 495872505, 41262262505, ...
%p A245397 b:= proc(n, i, k) option remember; `if`(n=0 or i=1, 1,
%p A245397       add(b(n-j, i-1, k)*binomial(n, j)^k, j=0..n))
%p A245397     end:
%p A245397 A:= (n, k)-> b(n$2, k):
%p A245397 seq(seq(A(n, d-n), n=0..d), d=0..10);
%t A245397 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_ *)
%Y A245397 Columns k=0-10 give: A001700(n-1) for n>0, A000312, A033935, A055733, A055740, A246240, A246241, A246242, A246243, A246244, A246245.
%Y A245397 Rows n=0+1, 2 give: A000012, A052548.
%Y A245397 Main diagonal gives A245398.
%K A245397 nonn,tabl
%O A245397 0,6
%A A245397 _Alois P. Heinz_, Jul 21 2014