A100534 Number of partitions of 2*n into parts of two kinds.

1, 5, 20, 65, 185, 481, 1165, 2665, 5822, 12230, 24842, 49010, 94235, 177087, 326015, 589128, 1046705, 1831065, 3157789, 5374390, 9035539, 15018300, 24697480, 40210481, 64854575, 103679156, 164363280, 258508230, 403531208, 625425005

Offset: 0

Author

N. J. A. Sloane, Nov 27 2004

Keywords

• nonn

Examples

			G.f.: 1 + 5*x + 20*x^2 + 65*x^3 + 185*x^4 + 481*x^5 + 1165*x^6 + 2665*x^7 + ...
G.f.: 1/q + 5*q^23 + 20*q^47 + 65*q^71 + 185*q^95 + 481*q^119 + 1165*q^143 + ...
a(1)=5 because we have 2, 2', 11, 1'1 and 1'1'.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000712, A100535.

Programs

Magma

m:=40;
f:= func< x | (&*[ (1-x^(4*n))^5/((1-x^n)^5*(1-x^(8*n))^2) : n in [1..m+2]]) >;
R:=PowerSeriesRing(Rationals(), m);
Coefficients(R!( f(x) )); // G. C. Greubel, Mar 27 2023

Maple

with(combinat): A000712:=n-> add(numbpart(k)*numbpart(n-k),k=0..n): seq(A000712(2*n),n=0..32); # Emeric Deutsch, Dec 16 2004

Mathematica

a[n_] := Sum[PartitionsP[k] PartitionsP[2 n - k], {k, 0, 2 n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 30 2015, adapted from Maple *)

PARI

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^5 / (eta(x + A)^5 * eta(x^8 + A)^2), n))} /* Michael Somos, Sep 24 2011 */

PARI

{a(n) = local(A); if( n<0, 0, n = 2*n; A = x * O(x^n); polcoeff( 1 / eta(x + A)^2, n))} /* Michael Somos, Sep 24 2011 */

SageMath

m=40
def f(x): return product( (1-x^(4*n))^5/((1-x^n)^5*(1-x^(8*n))^2) for n in range(1,m+2) )
def A100535_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).list()
A100535_list(m) # G. C. Greubel, Mar 27 2023

Formula

Expansion of q^(1/24) * eta(q^4)^5 / (eta(q)^5 * eta(q^8)^2) in powers of q. - Michael Somos, Sep 24 2011

a(n) = A000712(2*n).

Extensions

More terms from Emeric Deutsch, Dec 16 2004