

A117957


Number of partitions of n into parts larger than 1 and congruent to 1 mod 4.


3



1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 2, 1, 1, 1, 2, 2, 1, 2, 3, 3, 2, 2, 4, 4, 3, 3, 5, 6, 5, 4, 6, 8, 7, 6, 8, 10, 10, 9, 10, 13, 13, 12, 14, 17, 18, 16, 18, 22, 23, 22, 23, 28, 31, 29, 30, 36, 39, 39, 39, 45, 51, 50, 51, 57, 64, 65, 65, 73, 81, 83, 84, 91, 102, 106, 106
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OFFSET

0,19


COMMENTS

Also number of partitions of n such that 2k and 2k+1 occur with the same multiplicities. Example: a(26)=3 because we have [11,10,3,2], [9,8,5,4] and [7,7,6,6]. It is easy to find a bijection between these partitions and those described in the definition.


LINKS

Table of n, a(n) for n=0..82.


FORMULA

G.f.: 1/product(1x^(4i+1), i=1..infinity).
a(n) ~ exp(sqrt(n/6)*Pi) * Pi^(1/4) * Gamma(1/4) / (2^(31/8) * 3^(5/8) * n^(9/8)).  Vaclav Kotesovec, Mar 07 2016


EXAMPLE

a(26)=3 because we have [21,5],[17,9] and [13,13].


MAPLE

g:=1/product(1x^(4*i+1), i=1..50): gser:=series(g, x=0, 93): seq(coeff(gser, x, n), n=0..88);


MATHEMATICA

nmax = 100; CoefficientList[Series[Product[1/(1x^(4*k+1)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 07 2016 *)


CROSSREFS

Cf. A035451, A035462.
Sequence in context: A220492 A229873 A135230 * A145702 A145704 A139632
Adjacent sequences: A117954 A117955 A117956 * A117958 A117959 A117960


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Apr 05 2006


STATUS

approved



