A271619 Twice partitioned numbers where the first partition is strict.

1, 1, 2, 5, 8, 18, 34, 65, 109, 223, 386, 698, 1241, 2180, 3804, 6788, 11390, 19572, 34063, 56826, 96748, 163511, 272898, 452155, 755928, 1244732, 2054710, 3382147, 5534696, 8992209, 14733292, 23763685, 38430071, 62139578, 99735806, 160183001, 256682598

Offset

0,3

Comments

Number of sequences of integer partitions of the parts of some strict partition of n.

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1, g(n) = -A000041(n). - Seiichi Manyama, Nov 15 2018

Links

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

Formula

G.f.: Product_{i>=1} (1 + A000041(i) * x^i).

Example

a(6)=34: {(6);(5)(1),(51);(4)(2),(42);(4)(11),(41)(1),(411);(33);(3)(2)(1),(31)(2),(32)(1),(321);(3)(11)(1),(31)(11),(311)(1),(3111);(22)(2),(222);(21)(2)(1),(22)(11),(211)(2),(221)(1),(2211);(21)(11)(1),(111)(2)(1),(211)(11),(1111)(2),(2111)(1),(21111);(111)(11)(1),(1111)(11),(11111)(1),(111111)}

Maple

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,
       b(n-i, i-1)*combinat[numbpart](i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 11 2016

Mathematica

With[{n = 50}, CoefficientList[Series[Product[(1 + PartitionsP[i] x^i), {i, 1, n}], {x, 0, n}], x]]

Crossrefs

Cf. A000009, A000041, A063834 (twice partitioned numbers), A270995, A279785, A327552, A327607.

Sequence in context: A304966 A354539 A152006 * A197211 A256723 A032063

Adjacent sequences: A271616 A271617 A271618 * A271620 A271621 A271622

Keyword

nonn

Author

Gus Wiseman, Apr 10 2016

Status

approved