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%I #14 Dec 19 2025 08:37:27
%S 1,3,9,27,87,351,1539,6723,28467,117531,479763,1955907,7998075,
%T 32807835,134797203,554013027,2276268075,9348884811,38388273219,
%U 157620118995,647198010171,2657558176443,10912933399635,44813051604675,184020663786507,755659784907243
%N a(n) = Sum_{k=0..floor(n/4)} 2^k * 3^(n-3*k) * binomial(2*(n-3*k),2*k).
%H Seiichi Manyama, <a href="/A391757/b391757.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (6,-9,0,12,36,0,0,-36).
%F G.f.: (1-3*x-6*x^4) / ((1-3*x-6*x^4)^2 - 72*x^5).
%F a(n) = 6*a(n-1) - 9*a(n-2) + 12*a(n-4) + 36*a(n-5) - 36*a(n-8).
%o (PARI) my(A=2, B=3, C=4*A*B^2, N=1, M=30, x='x+O('x^M), X=1-B*x-A*B*x^4, Y=5); Vec(sum(k=0, N\2, C^k*binomial(N, 2*k)*X^(N-2*k)*x^(Y*k))/(X^2-C*x^Y)^N)
%Y Cf. A391724, A391758.
%K nonn
%O 0,2
%A _Seiichi Manyama_, Dec 18 2025