A035457 - OEIS

A035457

Number of partitions of n into parts of the form 4*k + 2.

1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, 8, 0, 10, 0, 12, 0, 15, 0, 18, 0, 22, 0, 27, 0, 32, 0, 38, 0, 46, 0, 54, 0, 64, 0, 76, 0, 89, 0, 104, 0, 122, 0, 142, 0, 165, 0, 192, 0, 222, 0, 256, 0, 296, 0, 340, 0, 390, 0, 448, 0, 512, 0, 585, 0, 668, 0, 760, 0, 864, 0, 982, 0

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OFFSET

0,7

COMMENTS

Also number of partitions of n into distinct even parts. Example: a(10)=3 because we have [10],[8,2] and [6,4]. - Emeric Deutsch, Feb 22 2006

Also number of partitions of n into odd parts, each part occurring an even number of times. Example: a(10)=3 because we have [5,5], [3,3,1,1,1,1] and [1,1,1,1,1,1,1,1,1,1]. - Emeric Deutsch, Apr 08 2006

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

FORMULA

G.f.: 1/Product_{n>=0} (1 - x^(4*n+2)).

G.f.: 1/Product_{j>=0} (1 - x^(8*j+2))*(1 - x^(8*j+6)).

G.f.: Product_{j>=1} (1 + x^(2*j)). - Emeric Deutsch, Feb 22 2006

a(2*n-1) = 0, a(2*n) = A000009(n). a(n) = A116675(n,0). - Emeric Deutsch, Feb 22 2006

G.f.: Sum_{n>=1} (x^(n*(n+1)) / Product_{k=1..n} (1 - x^(2*k))). - Joerg Arndt, Mar 10 2011

If n is even, a(n) ~ exp(Pi*sqrt(n/6)) / (2^(5/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Feb 26 2015

a(4*n) = A035294(n) and a(4*n+2) = A078408(n). - George Beck, Aug 19 2017

EXAMPLE

a(10)=3 because we have [10], [6,2,2] and [2,2,2,2,2].

MAPLE

g:=product(1+x^(2*j), j=1..45): gser:=series(g, x=0, 85): seq(coeff(gser, x, n), n=0..79); # Emeric Deutsch, Feb 22 2006; a(0) added by Georg Fischer, Dec 10 2020

MATHEMATICA

nn=80; CoefficientList[Series[Product[1+ x^(2i), {i, 1, nn}], {x, 0, nn}], x] (* Geoffrey Critzer, Jun 20 2014 *)

nmax = 50; kmax = nmax/4; s = Range[0, kmax]*4 + 2;

Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Aug 03 2020 *)

PROG

(PARI)

N=166; S=2+sqrtint(N); x='x+O('x^N);

gf=sum(n=0, S, x^(n^2+n)/prod(k=1, n, 1-x^(2*k)) );

Vec(gf)

\\ Joerg Arndt, Feb 18 2014

CROSSREFS

Cf. A000726, A116675.