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A100534 Number of partitions of 2*n into parts of two kinds. 1

%I #20 Mar 28 2023 08:00:55

%S 1,5,20,65,185,481,1165,2665,5822,12230,24842,49010,94235,177087,

%T 326015,589128,1046705,1831065,3157789,5374390,9035539,15018300,

%U 24697480,40210481,64854575,103679156,164363280,258508230,403531208,625425005

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

%H G. C. Greubel, <a href="/A100534/b100534.txt">Table of n, a(n) for n = 0..1000</a>

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

%e G.f.: 1 + 5*x + 20*x^2 + 65*x^3 + 185*x^4 + 481*x^5 + 1165*x^6 + 2665*x^7 + ...

%e G.f.: 1/q + 5*q^23 + 20*q^47 + 65*q^71 + 185*q^95 + 481*q^119 + 1165*q^143 + ...

%e a(1)=5 because we have 2, 2', 11, 1'1 and 1'1'.

%p 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 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 (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 (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 (Magma)

%o m:=40;

%o f:= func< x | (&*[ (1-x^(4*n))^5/((1-x^n)^5*(1-x^(8*n))^2) : 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 product( (1-x^(4*n))^5/((1-x^n)^5*(1-x^(8*n))^2) 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, A100535.

%K nonn

%O 0,2

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

%E More terms from _Emeric Deutsch_, Dec 16 2004

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Last modified March 28 04:13 EDT 2024. Contains 371235 sequences. (Running on oeis4.)