|
%I #44 Mar 03 2024 14:49:05
%S 1,1,2,5,12,29,69,165,393,937,2233,5322,12683,30227,72037,171680,
%T 409151,975097,2323870,5538294,13198973,31456058,74966710,178662171,
%U 425791279,1014754341,2418382956,5763538903,13735781840,32735391558,78015643589
%N Expansion of 1/( 1 - x / Product_{n>=1} (1-x^n) ).
%C Previous name was: Invert transform of right-shifted partition function (A000041).
%C Sums of the antidiagonals of the array formed by sequences A000007, A000041, A000712, A000716, ... or its transpose A000012, A000027, A000096, A006503, A006504, ....
%C 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
%C 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
%C From _L. Edson Jeffery_, Mar 16 2011: (Start)
%C 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)
%C (1, 0, ... )
%C (1, 1, 0, ... )
%C (2, 2, 1, 0, ... )
%C (3, 5, 3, 1, 0, ... )
%C (5, 10, 9, 4, 1, 0, ...)
%C etc., and a(n) is the sum of entries in row n of T. (End)
%H Alois P. Heinz, <a href="/A067687/b067687.txt">Table of n, a(n) for n = 0..1000</a>
%H Ricardo Gómez Aíza, <a href="https://arxiv.org/abs/2402.16111">Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis</a>, arXiv:2402.16111 [math.CO], 2024. See p. 23.
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F a(n) = Sum_{k=1..n} A000041(k-1)*a(n-k). - _Vladeta Jovovic_, Apr 07 2003
%F O.g.f.: 1/(1-x*P(x)), P(x) - o.g.f. for number of partitions (A000041). - _Vladimir Kruchinin_, Aug 10 2010
%F 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
%e The array begins:
%e 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 2, 3, 4, 5, 6, 7, ...
%e 0, 2, 5, 9, 14, 20, 27, ...
%e 0, 3, 10, 22, 40, 65, ...
%e 0, 5, 20, 51, 105, ...
%e 0, 7, 36, 108, ...
%e 0, 11, 65, ...
%o (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
%Y Cf. A000007, A000041, A000712, A000716, A000012, A000027, A000096, A006503, A006504.
%Y Cf. table A060850.
%Y Cf. A137682, A143866.
%Y Antidiagonal sums of A144064.
%K nonn
%O 0,3
%A _Alford Arnold_, Feb 05 2002
%E More terms from _Vladeta Jovovic_, Apr 07 2003
%E More terms and better definition from _Franklin T. Adams-Watters_, Mar 14 2006
%E New name (using g.f. by _Vladimir Kruchinin_), _Joerg Arndt_, Feb 19 2014
|
