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%I #28 Jul 10 2018 19:14:30
%S 1,1,0,1,1,0,1,2,3,0,1,3,7,6,0,1,4,12,18,13,0,1,5,18,37,47,24,0,1,6,
%T 25,64,111,110,48,0,1,7,33,100,215,303,258,86,0,1,8,42,146,370,660,
%U 804,568,160,0,1,9,52,203,588,1251,1938,2022,1237,282,0
%N Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is Euler transform of (j->j*k).
%C A(n,k) is the number of partitions of n when parts i are of k*i kinds. A(2,2) = 7: [2a], [2b], [2c], [2d], [1a,1a], [1a,1b], [1b,1b].
%H Alois P. Heinz, <a href="/A255961/b255961.txt">Rows n = 0..140, flattened</a>
%F G.f. of column k: Product_{j>=1} 1/(1-x^j)^(j*k).
%F T(n,k) = Sum_{i=0..k} C(k,i) * A257673(n,k-i).
%e Square array A(n,k) begins:
%e 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 2, 3, 4, 5, 6, 7, ...
%e 0, 3, 7, 12, 18, 25, 33, 42, ...
%e 0, 6, 18, 37, 64, 100, 146, 203, ...
%e 0, 13, 47, 111, 215, 370, 588, 882, ...
%e 0, 24, 110, 303, 660, 1251, 2160, 3486, ...
%e 0, 48, 258, 804, 1938, 4005, 7459, 12880, ...
%e 0, 86, 568, 2022, 5400, 12150, 24354, 44885, ...
%p A:= proc(n, k) option remember; `if`(n=0, 1, k*add(
%p A(n-j, k)*numtheory[sigma][2](j), j=1..n)/n)
%p end:
%p seq(seq(A(n, d-n), n=0..d), d=0..12);
%t A[n_, k_] := A[n, k] = If[n==0, 1, k*Sum[A[n-j, k]*DivisorSigma[2, j], {j, 1, n}]/n]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* _Jean-François Alcover_, Feb 02 2016, after _Alois P. Heinz_ *)
%Y Columns k=0-10 give: A000007, A000219, A161870, A255610, A255611, A255612, A255613, A255614, A193427, A316461, A316462.
%Y Rows n=0-3 give: A000012, A001477, A055998, A101853.
%Y Main diagonal gives A255672.
%Y Antidiagonal sums give A299166.
%Y Cf. A144064, A257673.
%K nonn,tabl
%O 0,8
%A _Alois P. Heinz_, Mar 11 2015
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