A392032 - OEIS

%I #14 Dec 31 2025 14:26:13

%S 1,1,7,19,73,241,831,2787,9339,31003,102373,336073,1097911,3570607,

%T 11565529,37324437,120052107,384962859,1230978093,3926097681,

%U 12492150819,39660238779,125656218933,397360559841,1254333690477,3952929646701,12437932152051

%N a(n) = Sum_{k=0..floor(n/2)} binomial(k+2,2) * binomial(n,k) * binomial(n-k,k).

%H Seiichi Manyama, <a href="/A392032/b392032.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: ((1-x)^4 - 4*x^2*(1-x)^2 + 6*x^4) / ((1-x)^2 - 4*x^2)^(5/2).

%t CoefficientList[Series[((1-x)^4-4*x^2*(1-x)^2+6*x^4)/((1-x)^2-4*x^2)^(5/2),{x,0,50}],x] (* _Vincenzo Librandi_, Dec 31 2025 *)

%o (PARI) a098473(n, k) = binomial(n, k)*binomial(2*k, k);

%o my(A=1, B=1, C=A*B, N=2, M=30, x='x+O('x^M), X=1-B*x, Y=2); Vec(sum(k=0, N, (-C)^k*a098473(N, k)*X^(2*N-2*k)*x^(Y*k))/(X^2-4*C*x^Y)^(N+1/2))

%o (Magma) m := 50; R<x> := PowerSeriesRing(RationalField(), m); Coefficients(((1-x)^4 - 4*x^2*(1-x)^2 + 6*x^4) / ((1-x)^2 - 4*x^2)^(5/2)); // _Vincenzo Librandi_, Dec 31 2025

%Y Cf. A002426, A391990, A392033.

%Y Cf. A098473.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Dec 27 2025