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A255961 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is Euler transform of (j->j*k).

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