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

%I A100534 #20 Mar 28 2023 08:00:55
%S A100534 1,5,20,65,185,481,1165,2665,5822,12230,24842,49010,94235,177087,
%T A100534 326015,589128,1046705,1831065,3157789,5374390,9035539,15018300,
%U A100534 24697480,40210481,64854575,103679156,164363280,258508230,403531208,625425005
%N A100534 Number of partitions of 2*n into parts of two kinds.
%H A100534 G. C. Greubel, <a href="/A100534/b100534.txt">Table of n, a(n) for n = 0..1000</a>
%F A100534 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
%F A100534 a(n) = A000712(2*n).
%e A100534 G.f.: 1 + 5*x + 20*x^2 + 65*x^3 + 185*x^4 + 481*x^5 + 1165*x^6 + 2665*x^7 + ...
%e A100534 G.f.: 1/q + 5*q^23 + 20*q^47 + 65*q^71 + 185*q^95 + 481*q^119 + 1165*q^143 + ...
%e A100534 a(1)=5 because we have 2, 2', 11, 1'1 and 1'1'.
%p A100534 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
%t A100534 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 *)
%o A100534 (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 */
%o A100534 (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 */
%o A100534 (Magma)
%o A100534 m:=40;
%o A100534 f:= func< x | (&*[ (1-x^(4*n))^5/((1-x^n)^5*(1-x^(8*n))^2) : n in [1..m+2]]) >;
%o A100534 R<x>:=PowerSeriesRing(Rationals(), m);
%o A100534 Coefficients(R!( f(x) )); // _G. C. Greubel_, Mar 27 2023
%o A100534 (SageMath)
%o A100534 m=40
%o A100534 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) )
%o A100534 def A100535_list(prec):
%o A100534     P.<x> = PowerSeriesRing(QQ, prec)
%o A100534     return P( f(x) ).list()
%o A100534 A100535_list(m) # _G. C. Greubel_, Mar 27 2023
%Y A100534 Cf. A000712, A100535.
%K A100534 nonn
%O A100534 0,2
%A A100534 _N. J. A. Sloane_, Nov 27 2004
%E A100534 More terms from _Emeric Deutsch_, Dec 16 2004