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 A067687 Expansion of 1/( 1 - x / Product_{n>=1} (1-x^n) ). 19

%I

%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)=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 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 = 0.41960035259835647849877575356670002531808936312... 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

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Last modified June 20 19:36 EDT 2019. Contains 324234 sequences. (Running on oeis4.)