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A067687
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Invert transform of right-shifted partition function (A000041).
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2
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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; internal format)
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OFFSET
| 0,3
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COMMENTS
| 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,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 04 2008]
Starting with offset 1 = A137682 / A000041; i.e. (1, 3, 7, 17, 40, 96,...) / (1, 2, 3, 5, 7, 11,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 01 2009]
From L. Edson Jeffery, March 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)
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LINKS
| N. J. A. Sloane, Transforms
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FORMULA
| a(n) = Sum_{k=1..n} A000041(k-1)*a(n-k). - Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 07 2003
O.g.f. 1/(1-x*P(x)), P(x) - o.g.f. for number of partitions (A000041) [From Kruchinin Vladimir (kru(AT)ie.tusur.ru), Aug 10 2010]
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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 ...
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PROG
| (From Joerg Arndt, May 08 2009) x='x+O('x^55) v=Vec( Ser( sum(n=0, 33, x^(n)/eta(x)^n ) ) )
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CROSSREFS
| Cf. A000007, A000041, A000712, A000716, A000012, A000027, A000096, A006503, A006504.
Cf. table A060850.
Cf, A137682, A143866.
Sequence in context: A131045 A026721 A094975 * A130009 A048624 A176981
Adjacent sequences: A067684 A067685 A067686 * A067688 A067689 A067690
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KEYWORD
| nonn
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AUTHOR
| Alford Arnold (Alford1940(AT)AOL.COM), Feb 05 2002
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EXTENSIONS
| More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 07 2003
More terms and better definition from Frank Adams-Watters (FrankTAW(AT)Netscape.net), Mar 14 2006
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