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A061260 G.f.: Product_{k>=1} (1-y*x^k)^(-numbpart(k)), where numbpart(k) = number of partitions of k, cf. A000041.

%I A061260 #26 Feb 17 2023 21:37:30
%S A061260 1,2,1,3,2,1,5,6,2,1,7,11,6,2,1,11,23,15,6,2,1,15,40,32,15,6,2,1,22,
%T A061260 73,67,37,15,6,2,1,30,120,134,79,37,15,6,2,1,42,202,255,172,85,37,15,
%U A061260 6,2,1,56,320,470,348,187,85,37,15,6,2,1,77,511,848,697,397,194,85,37,15,6,2,1
%N A061260 G.f.: Product_{k>=1} (1-y*x^k)^(-numbpart(k)), where numbpart(k) = number of partitions of k, cf. A000041.
%C A061260 Multiset transformation of A000041. - _R. J. Mathar_, Apr 30 2017
%C A061260 Number of orderless twice-partitions of n of length k. A twice-partition of n is a choice of a partition of each part in a partition of n. The T(5,3) = 6 orderless twice-partitions: (3)(1)(1), (21)(1)(1), (111)(1)(1), (2)(2)(1), (2)(11)(1), (11)(11)(1). - _Gus Wiseman_, Mar 23 2018
%H A061260 Alois P. Heinz, <a href="/A061260/b061260.txt">Rows n = 1..141, flattened</a>
%H A061260 <a href="/index/Mu#multiplicative_completely">Index entries for triangles generated by the Multiset Transformation</a>
%e A061260 :  1;
%e A061260 :  2,   1;
%e A061260 :  3,   2,   1;
%e A061260 :  5,   6,   2,   1;
%e A061260 :  7,  11,   6,   2,  1;
%e A061260 : 11,  23,  15,   6,  2,  1;
%e A061260 : 15,  40,  32,  15,  6,  2,  1;
%e A061260 : 22,  73,  67,  37, 15,  6,  2, 1;
%e A061260 : 30, 120, 134,  79, 37, 15,  6, 2, 1;
%e A061260 : 42, 202, 255, 172, 85, 37, 15, 6, 2, 1;
%p A061260 b:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
%p A061260       `if`(min(i, p)<1, 0, add(b(n-i*j, i-1, p-j)*binomial(
%p A061260        combinat[numbpart](i)+j-1, j), j=0..min(n/i, p)))))
%p A061260     end:
%p A061260 T:= (n, k)-> b(n$2, k):
%p A061260 seq(seq(T(n, k), k=1..n), n=1..14);  # _Alois P. Heinz_, Apr 13 2017
%t A061260 b[n_, i_, p_] := b[n, i, p] = If[p > n, 0, If[n == 0, 1, If[Min[i, p] < 1, 0, Sum[b[n - i*j, i - 1, p - j]*Binomial[PartitionsP[i] + j - 1, j], {j, 0, Min[n/i, p]}]]]];
%t A061260 T[n_, k_] := b[n, n, k];
%t A061260 Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, May 17 2018, after _Alois P. Heinz_ *)
%Y A061260 Row sums: A001970, first column: A000041.
%Y A061260 T(2,n) gives A061261,
%Y A061260 Cf. A063834, A119442, A273873, A285229, A289078, A289501, A299200, A299201.
%K A061260 easy,nonn,tabl
%O A061260 1,2
%A A061260 _Vladeta Jovovic_, Apr 23 2001