A067687 Expansion of 1/( 1 - x / Product_{n>=1} (1-x^n) ).

1, 1, 2, 5, 12, 29, 69, 165, 393, 937, 2233, 5322, 12683, 30227, 72037, 171680, 409151, 975097, 2323870, 5538294, 13198973, 31456058, 74966710, 178662171, 425791279, 1014754341, 2418382956, 5763538903, 13735781840, 32735391558, 78015643589

Offset: 0

Author

Alford Arnold, Feb 05 2002

Keywords

• nonn

Comments

Previous name was: Invert transform of right-shifted partition function (A000041).

Sums of the antidiagonals of the array formed by sequences A000007, A000041, A000712, A000716, ... or its transpose A000012, A000027, A000096, A006503, A006504, ....

Row sums of triangle A143866 = (1, 2, 5, 12, 29, 69, 165, ...) and right border of A143866 = (1, 1, 2, 5, 12, ...). - Gary W. Adamson, Sep 04 2008

Starting with offset 1 = A137682 / A000041; i.e. (1, 3, 7, 17, 40, 96, ...) / (1, 2, 3, 5, 7, 11, ...). - Gary W. Adamson, May 01 2009

From L. Edson Jeffery, Mar 16 2011: (Start)

Another approach is the following. Let T be the infinite lower triangular matrix with columns C_k (k=0,1,2,...) such that C_0=A000041 and, for k > 0, such that C_k is the sequence giving the number of partitions of n into parts of k+1 kinds (successive self-convolutions of A000041 yielding A000712, A000716, ...) and shifted down by k rows. Then T begins (ignoring trailing zero entries in the rows)

(1, 0, ... )

(1, 1, 0, ... )

(2, 2, 1, 0, ... )

(3, 5, 3, 1, 0, ... )

(5, 10, 9, 4, 1, 0, ...)

etc., and a(n) is the sum of entries in row n of T. (End)

Examples

			The array begins:
  1,  1,  1,   1,   1,  1,  1, 1, ...
  0,  1,  2,   3,   4,  5,  6, 7, ...
  0,  2,  5,   9,  14, 20, 27, ...
  0,  3, 10,  22,  40, 65, ...
  0,  5, 20,  51, 105, ...
  0,  7, 36, 108, ...
  0, 11, 65, ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 23.
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000007, A000041, A000712, A000716, A000012, A000027, A000096, A006503, A006504.

Cf. table A060850.

Cf. A137682, A143866.

Antidiagonal sums of A144064.

Programs

PARI

N=66; x='x+O('x^N); et=eta(x); Vec( sum(n=0,N, x^n/et^n ) ) \\ Joerg Arndt, May 08 2009

Formula

a(n) = Sum_{k=1..n} A000041(k-1)*a(n-k). - Vladeta Jovovic, Apr 07 2003

O.g.f.: 1/(1-x*P(x)), P(x) - o.g.f. for number of partitions (A000041). - Vladimir Kruchinin, Aug 10 2010

a(n) ~ c / r^n, where r = A347968 = 0.419600352598356478498775753566700025318... is the root of the equation QPochhammer(r) = r and c = 0.3777957165566422058901624844315414446044096308877617181754... = Log[r]/(Log[(1 - r)*r] + QPolyGamma[1, r] - Log[r]*Derivative[0, 1][QPochhammer][r, r]). - Vaclav Kotesovec, Feb 16 2017, updated Mar 31 2018

Extensions

More terms from Vladeta Jovovic, Apr 07 2003

More terms and better definition from Franklin T. Adams-Watters, Mar 14 2006

New name (using g.f. by Vladimir Kruchinin), Joerg Arndt, Feb 19 2014