A067687 - OEIS

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 (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) is the sum of entries in row n of T. (End)`
	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`
	FORMULA	`a(n) = Sum_{k=1..n} A000041(k-1)a(n-k). - Vladeta Jovovic, Apr 07 2003` `O.g.f.: 1/(1-xP(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`
	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 * A291235 A130009 A324979` `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`