A100535 Number of partitions of 2*n + 1 into parts of two kinds.

2, 10, 36, 110, 300, 752, 1770, 3956, 8470, 17490, 35002, 68150, 129512, 240840, 439190, 786814, 1386930, 2408658, 4126070, 6978730, 11664896, 19283830, 31551450, 51124970, 82088400, 130673928, 206327710, 323275512, 502810130

Offset

0,1

Links

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

Formula

Expansion of q^(-11/24) * 2 * eta(q^2)^2 * eta(q^8)^2 / (eta(q)^5 * eta(q^4)) In powers of q. - Michael Somos, Sep 24 2011

a(n) = A000712(2*n + 1).

Example

G.f.: 2 + 10*x + 36*x^2 + 110*x^3 + 300*x^4 + 752*x^5 + 1770*x^6 + 3956*x^7 + ...
G.f.: 2*q^11 + 10*q^35 + 36*q^59 + 110*q^83 + 300*q^107 + 752*q^131 + 1770*q^155 + ...
a(1)=10 because we have 3, 3', 21, 2'1, 21', 2'1', 111, 1'11, 1'1'1, 1'1'1'.

Maple

with(combinat): A000712:=n->sum(numbpart(k)*numbpart(n-k), k=0..n): seq(A000712(2*n-1), n=1..32); # Emeric Deutsch, Dec 16 2004

Mathematica

a[n_]:= Sum[PartitionsP[k] PartitionsP[2n+1-k], {k, 0, 2n+1}];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 30 2015, adapted from Maple *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( 2 * eta(x^2 + A)^2 * eta(x^8 + A)^2 / (eta(x + A)^5 * eta(x^4 + A)), n))} /* Michael Somos, Sep 24 2011 */
(PARI) {a(n) = local(A); if( n<0, 0, n = 2*n + 1; A = x * O(x^n); polcoeff( 1 / eta(x + A)^2, n))} /* Michael Somos, Sep 24 2011 */
(Magma)
m:=40;
f:= func< x | 2*(&*[ ((1-x^(2*n))^2*(1-x^(8*n))^2)/((1-x^n)^5*(1-x^(4*n))) : n in [1..m+2]]) >;
R<x>:=PowerSeriesRing(Rationals(), m);
Coefficients(R!( f(x) )); // G. C. Greubel, Mar 27 2023
(SageMath)
m=40
def f(x): return 2*product( ((1-x^(2*n))^2*(1-x^(8*n))^2)/((1-x^n)^5*(1-x^(4*n))) for n in range(1, m+2) )
def A100535_list(prec):
    P.<x> = PowerSeriesRing(QQ, prec)
    return P( f(x) ).list()
A100535_list(m) # G. C. Greubel, Mar 27 2023

Crossrefs

Cf. A000712.

Sequence in context: A317454 A244715 A212573 * A340885 A206622 A266942

Adjacent sequences: A100532 A100533 A100534 * A100536 A100537 A100538

Keyword

nonn

Author

N. J. A. Sloane, Nov 27 2004

Extensions

More terms from Emeric Deutsch, Dec 16 2004

Status

approved