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A100535
Number of partitions of 2*n + 1 into parts of two kinds.
2
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
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
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Nov 27 2004
EXTENSIONS
More terms from Emeric Deutsch, Dec 16 2004
STATUS
approved