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A035462 Number of partitions of n into parts 4k-1. 7
1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 1, 2, 3, 2, 2, 4, 4, 3, 4, 5, 5, 5, 6, 7, 8, 7, 8, 11, 10, 10, 13, 14, 14, 15, 17, 19, 20, 20, 24, 27, 26, 28, 33, 35, 35, 39, 44, 46, 48, 52, 58, 62, 63, 69, 78, 80, 83, 93, 100, 104, 111, 120, 130, 137, 143, 156, 169, 175, 185, 203 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,15

COMMENTS

Also, number of partitions into parts 8k+3 or 8k+7.

Also number of partitions of n such that 2k-1 and 2k occur with the same multiplicity. Example: a(18)=3 because we have [8,7,2,1],[6,5,4,3] and [2,2,2,2,2,2,1,1,1,1,1,1]. It is easy to find a bijection between these partitions and those described in the definition. - Emeric Deutsch, Apr 05 2006

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

FORMULA

G.f.: 1/prod(i>=1, 1-x^(4*i-1)). - Emeric Deutsch, Apr 05 2006

G.f.: sum(n>=0, x^(3*n) / prod(k=1..n, 1-x^(4*k) ) ) = 1 + sum(n>=0, x^(4*n+3) / prod(k>=n, 1-x^(4*k+3) ) ) = 1 + sum(n>=0, x^(4*n+3) / prod(k=0..n, 1-x^(4*k+3) ) ). - Joerg Arndt, Apr 08 2011

a(n) ~ Pi^(3/4) * exp(Pi*sqrt(n/6)) / (Gamma(1/4) * 2^(13/8) * 3^(3/8) * n^(7/8)) * (1 + (Pi/(96*sqrt(6)) - 21*sqrt(3/2)/(16*Pi)) / sqrt(n)). - Vaclav Kotesovec, Feb 26 2015, extended Jan 24 2017

a(n) = (1/n)*Sum_{k=1..n} A050452(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 20 2017

EXAMPLE

a(18)=3 because we have [15,3],[11,7] and [3,3,3,3,3,3].

MAPLE

g:=1/product(1-x^(4*i-1), i=1..50): gser:=series(g, x=0, 80): seq(coeff(gser, x, n), n=1..75); # Emeric Deutsch, Apr 05 2006

MATHEMATICA

nmax = 100; CoefficientList[Series[Product[1/(1-x^(4*k+3)), {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 26 2015 *)

CROSSREFS

Cf. A035441-A035468, A050452.

Cf. similar sequences of number of partitions of n into parts congruent to m-1 mod m: A000009 (m=2), A035386 (m=3), this sequence (m=4), A109700 (m=5), A109702 (m=6), A109708 (m=7).

Sequence in context: A237284 A294186 A294185 * A260414 A160735 A216338

Adjacent sequences:  A035459 A035460 A035461 * A035463 A035464 A035465

KEYWORD

nonn

AUTHOR

Olivier Gérard

EXTENSIONS

Offset changed by N. J. A. Sloane, Apr 11 2010

STATUS

approved

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Last modified April 20 22:22 EDT 2019. Contains 322310 sequences. (Running on oeis4.)