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 A067687 Expansion of 1/( 1 - x / Product_{n>=1} (1-x^n) ). 19
 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... = 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 A048624 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 December 17 00:23 EST 2018. Contains 318191 sequences. (Running on oeis4.)