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A067687 Expansion of 1/( 1 - x / Product_{n>=1} (1-x^n) ). 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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)=sum of entries in row n of T. (End)

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

N. J. A. Sloane, Transforms

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 = 0.41960035259835647849877575356670002531808936312... is the root of the equation QPochhammer(r) = r and c = 0.3777957165566422058901624844315414446044096308877617181754... . - Vaclav Kotesovec, Feb 16 2017

EXAMPLE

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 ...

PROG

(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

CROSSREFS

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

Cf. table A060850.

Cf. A137682, A143866.

Antidiagonal sums of A144064.

Sequence in context: A182555 A026721 A094975 * A130009 A048624 A176981

Adjacent sequences:  A067684 A067685 A067686 * A067688 A067689 A067690

KEYWORD

nonn

AUTHOR

Alford Arnold, Feb 05 2002

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

STATUS

approved

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Last modified May 23 23:53 EDT 2017. Contains 286937 sequences.