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