A117957 Number of partitions of n into parts larger than 1 and congruent to 1 mod 4.

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

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(1-x^(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(1-x^(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/(1-x^(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