A100535 - OEIS

%I #18 Mar 28 2023 08:00:26

%S 2,10,36,110,300,752,1770,3956,8470,17490,35002,68150,129512,240840,

%T 439190,786814,1386930,2408658,4126070,6978730,11664896,19283830,

%U 31551450,51124970,82088400,130673928,206327710,323275512,502810130

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

%H G. C. Greubel, <a href="/A100535/b100535.txt">Table of n, a(n) for n = 0..1000</a>

%F 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

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

%e G.f.: 2 + 10*x + 36*x^2 + 110*x^3 + 300*x^4 + 752*x^5 + 1770*x^6 + 3956*x^7 + ...

%e G.f.: 2*q^11 + 10*q^35 + 36*q^59 + 110*q^83 + 300*q^107 + 752*q^131 + 1770*q^155 + ...

%e a(1)=10 because we have 3, 3', 21, 2'1, 21', 2'1', 111, 1'11, 1'1'1, 1'1'1'.

%p 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

%t a[n_]:= Sum[PartitionsP[k] PartitionsP[2n+1-k], {k,0,2n+1}];

%t Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Nov 30 2015, adapted from Maple *)

%o (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 */

%o (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 */

%o (Magma)

%o m:=40;

%o 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]]) >;

%o R<x>:=PowerSeriesRing(Rationals(), m);

%o Coefficients(R!( f(x) )); // _G. C. Greubel_, Mar 27 2023

%o (SageMath)

%o m=40

%o 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) )

%o def A100535_list(prec):

%o     P.<x> = PowerSeriesRing(QQ, prec)

%o     return P( f(x) ).list()

%o A100535_list(m) # _G. C. Greubel_, Mar 27 2023

%Y Cf. A000712.

%K nonn

%O 0,1

%A _N. J. A. Sloane_, Nov 27 2004

%E More terms from _Emeric Deutsch_, Dec 16 2004