A279785 Number of ways to choose a strict partition of each part of a strict partition of n.

1, 1, 1, 3, 4, 7, 11, 18, 28, 47, 71, 108, 166, 252, 382, 587, 869, 1282, 1938, 2832, 4153, 6148, 8962, 12965, 18913, 27301, 39380, 56747, 81226, 115907, 166358, 236000, 334647, 475517, 671806, 947552, 1335679, 1875175, 2630584, 3687589, 5150585, 7183548

Offset

0,4

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1, g(n) = -A000009(n). - Seiichi Manyama, Nov 14 2018

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..5000 from Alois P. Heinz)

Gus Wiseman, Sequences enumerating triangles of integer partitions

Formula

G.f.: Product_{k>0} (1 + A000009(k)*x^k). - Seiichi Manyama, Nov 14 2018

Example

The a(6)=11 twice-partitions are:
((6)), ((5)(1)), ((51)), ((4)(2)), ((42)), ((41)(1)),
((3)(2)(1)), ((31)(2)), ((32)(1)), ((321)), ((21)(2)(1)).

Maple

with(numtheory):
g:= proc(n) option remember; `if`(n=0, 1, add(add(
      `if`(d::odd, d, 0), d=divisors(j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0,
      `if`(n=0, 1, b(n, i-1)+`if`(i>n, 0, g(i)*b(n-i, i-1))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..70);  # Alois P. Heinz, Dec 20 2016

Mathematica

nn=20; CoefficientList[Series[Product[(1+PartitionsQ[k]x^k), {k, nn}], {x, 0, nn}], x]
(* Second program: *)
g[n_] := g[n] = If[n==0, 1, Sum[Sum[If[OddQ[d], d, 0], {d, Divisors[j]}]* g[n - j], {j, 1, n}]/n]; b[n_, i_] := b[n, i] = If[n > i*(i + 1)/2, 0, If[n==0, 1, b[n, i-1] + If[i>n, 0, g[i]*b[n-i, i-1]]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Feb 07 2017, after Alois P. Heinz *)

Crossrefs

Cf. A000009, A063834, A270995, A279375, A327553, A327605.

Sequence in context: A147869 A319106 A250296 * A100581 A093090 A193686

Adjacent sequences: A279782 A279783 A279784 * A279786 A279787 A279788

Keyword

nonn

Author

Gus Wiseman, Dec 18 2016

Status

approved