A392033 - OEIS

%I #15 Dec 31 2025 12:34:33

%S 1,1,9,25,109,381,1421,5069,18075,63499,221323,764523,2621587,8928115,

%T 30220347,101721707,340659227,1135543323,3769023795,12460643043,

%U 41045661279,134748546639,440974812831,1438893866655,4682226213021,15197079464301,49206276341901

%N a(n) = Sum_{k=0..floor(n/2)} binomial(k+3,3) * binomial(n,k) * binomial(n-k,k).

%H Seiichi Manyama, <a href="/A392033/b392033.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: ((1-x)^6 - 6*x^2*(1-x)^4 + 18*x^4*(1-x)^2 - 20*x^6) / ((1-x)^2 - 4*x^2)^(7/2).

%t CoefficientList[Series[((1-x)^6-6*x^2*(1-x)^4+18*x^4*(1-x)^2-20*x^6)/((1-x)^2-4*x^2)^(7/2),{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=3, 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)^6 - 6*x^2*(1-x)^4 + 18*x^4*(1-x)^2 - 20*x^6) / ((1-x)^2 - 4*x^2)^(7/2)); // _Vincenzo Librandi_, Dec 31 2025

%Y Cf. A002426, A391990, A392032.

%Y Cf. A098473.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Dec 27 2025