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

%I A100535 #18 Mar 28 2023 08:00:26
%S A100535 2,10,36,110,300,752,1770,3956,8470,17490,35002,68150,129512,240840,
%T A100535 439190,786814,1386930,2408658,4126070,6978730,11664896,19283830,
%U A100535 31551450,51124970,82088400,130673928,206327710,323275512,502810130
%N A100535 Number of partitions of 2*n + 1 into parts of two kinds.
%H A100535 G. C. Greubel, <a href="/A100535/b100535.txt">Table of n, a(n) for n = 0..1000</a>
%F A100535 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 A100535 a(n) = A000712(2*n + 1).
%e A100535 G.f.: 2 + 10*x + 36*x^2 + 110*x^3 + 300*x^4 + 752*x^5 + 1770*x^6 + 3956*x^7 + ...
%e A100535 G.f.: 2*q^11 + 10*q^35 + 36*q^59 + 110*q^83 + 300*q^107 + 752*q^131 + 1770*q^155 + ...
%e A100535 a(1)=10 because we have 3, 3', 21, 2'1, 21', 2'1', 111, 1'11, 1'1'1, 1'1'1'.
%p A100535 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 A100535 a[n_]:= Sum[PartitionsP[k] PartitionsP[2n+1-k], {k,0,2n+1}];
%t A100535 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Nov 30 2015, adapted from Maple *)
%o A100535 (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 A100535 (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 A100535 (Magma)
%o A100535 m:=40;
%o A100535 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 A100535 R<x>:=PowerSeriesRing(Rationals(), m);
%o A100535 Coefficients(R!( f(x) )); // _G. C. Greubel_, Mar 27 2023
%o A100535 (SageMath)
%o A100535 m=40
%o A100535 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 A100535 def A100535_list(prec):
%o A100535     P.<x> = PowerSeriesRing(QQ, prec)
%o A100535     return P( f(x) ).list()
%o A100535 A100535_list(m) # _G. C. Greubel_, Mar 27 2023
%Y A100535 Cf. A000712.
%K A100535 nonn
%O A100535 0,1
%A A100535 _N. J. A. Sloane_, Nov 27 2004
%E A100535 More terms from _Emeric Deutsch_, Dec 16 2004