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

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, 73, 67, 37, 15, 6, 2, 1, 30, 120, 134, 79, 37, 15, 6, 2, 1, 42, 202, 255, 172, 85, 37, 15, 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

Offset

1,2

Comments

Multiset transformation of A000041. - R. J. Mathar, Apr 30 2017

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

Links

Alois P. Heinz, Rows n = 1..141, flattened

Index entries for triangles generated by the Multiset Transformation

Example

:  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,  73,  67,  37, 15,  6,  2, 1;
: 30, 120, 134,  79, 37, 15,  6, 2, 1;
: 42, 202, 255, 172, 85, 37, 15, 6, 2, 1;

Maple

b:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
      `if`(min(i, p)<1, 0, add(b(n-i*j, i-1, p-j)*binomial(
       combinat[numbpart](i)+j-1, j), j=0..min(n/i, p)))))
    end:
T:= (n, k)-> b(n$2, k):
seq(seq(T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Apr 13 2017

Mathematica

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[n_, k_] := b[n, n, k];
Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, May 17 2018, after Alois P. Heinz *)

Crossrefs

Row sums: A001970, first column: A000041.

T(2,n) gives A061261,

Cf. A063834, A119442, A273873, A285229, A289078, A289501, A299200, A299201.

Sequence in context: A104446 A131345 A134423 * A152097 A119442 A064861

Adjacent sequences: A061257 A061258 A061259 * A061261 A061262 A061263

Keyword

easy,nonn,tabl

Author

Vladeta Jovovic, Apr 23 2001

Status

approved