A103265 Number of partitions of n in which both even and odd square parts occur in 2 forms c, c* and with multiplicity 1. There no restriction on parts which are twice squares.

1, 2, 2, 2, 4, 6, 6, 6, 8, 12, 14, 14, 16, 22, 26, 26, 30, 38, 44, 46, 52, 62, 70, 74, 80, 96, 110, 116, 124, 146, 166, 174, 186, 210, 238, 254, 272, 302, 338, 362, 384, 426, 470, 502, 532, 588, 646, 686, 726, 792, 872, 926, 980, 1062

Offset

0,2

Comments

Convolution of A001156 and A033461. - Vaclav Kotesovec, Aug 18 2015

Links

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

Formula

G.f.: Product_{k>0}((1+x^k^2)/(1-x^k^2)).

a(n) ~ exp(3 * ((4-sqrt(2))*zeta(3/2))^(2/3) * Pi^(1/3) * n^(1/3) / 4) * ((4-sqrt(2))*zeta(3/2))^(2/3) / (2^(7/2) * sqrt(3) * Pi^(7/6) * n^(7/6)). - Vaclav Kotesovec, Dec 29 2016

Example

E.g. a(8)=8 because 8 can be written as 8, 44*, 422, 4*22, 4211*, 4*211*, 2222, 22211*.

Maple

series(product((1+x^(k^2))/(1-x^(k^2)), k=1..100), x=0, 100);

Mathematica

nmax = 50; CoefficientList[Series[Product[(1+x^(k^2)) / (1-x^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 18 2015 *)

Crossrefs

Cf. A001156, A015128, A033461, A280263, A279227, A306147.

Sequence in context: A378992 A182539 A170887 * A341695 A008238 A218870

Adjacent sequences: A103262 A103263 A103264 * A103266 A103267 A103268

Keyword

easy,nonn

Author

Noureddine Chair, Feb 27 2005

Status

approved