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A035451 Number of partitions of n into parts congruent to 1 mod 4. 13
1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 5, 6, 7, 7, 8, 10, 11, 12, 13, 15, 17, 18, 20, 23, 26, 28, 30, 34, 38, 41, 44, 49, 55, 60, 64, 70, 78, 85, 91, 99, 109, 119, 128, 138, 151, 164, 176, 190, 207, 225, 241, 259, 281, 304, 326, 349, 377, 408, 437, 467, 503, 542, 581 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

LINKS

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

FORMULA

G.f.: 1/prod(k>=0, 1-x^(4*k+1)). - Vladeta Jovovic, Nov 22 2002

G.f.: sum(n>=0, x^n / prod(k=1..n, 1-x^(4*k) ) ). [Joerg Arndt, Apr 07 2011]

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

a(n) ~ Gamma(1/4) * exp(Pi*sqrt(n/6)) / (2^(19/8) * 3^(1/8) * n^(5/8) * Pi^(3/4)) * (1 + (Pi/(96*sqrt(6)) - 5*sqrt(3/2)/(16*Pi)) / sqrt(n)). - Vaclav Kotesovec, Feb 26 2015, extended Jan 24 2017

a(n) = (1/n)*Sum_{k=1..n} A050449(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 20 2017

MATHEMATICA

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

CROSSREFS

Cf. A035462, A035382, A050449.

Cf. similar sequences of number of partitions of n into parts congruent to 1 mod m: A000009 (m=2), A035382 (m=3), this sequence (m=4), A109697 (m=5), A109701 (m=6), A109703 (m=7), A277090 (m=8).

Sequence in context: A280168 A001156 A199119 * A124746 A124789 A103372

Adjacent sequences:  A035448 A035449 A035450 * A035452 A035453 A035454

KEYWORD

nonn

AUTHOR

Olivier Gérard

EXTENSIONS

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

STATUS

approved

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Last modified February 20 22:25 EST 2018. Contains 299387 sequences. (Running on oeis4.)