A391758 - OEIS

%I #13 Dec 19 2025 08:37:15

%S 1,6,27,108,417,1782,8343,40176,190377,876906,3947211,17522244,

%T 77243517,339290046,1486173663,6490123704,28248529761,122547210930,

%U 530018594883,2286202048812,9838251072261,42248728829286,181086392267559,774818305073568,3309874281409257

%N a(n) = Sum_{k=0..floor(n/4)} (n-3*k+1) * 2^k * 3^(n-3*k) * binomial(2*(n-3*k),2*k).

%H Seiichi Manyama, <a href="/A391758/b391758.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (12,-54,108,-57,-72,-216,648,-216,-432,-1944,0,864,2592,0,0,-1296).

%F G.f.: ((1-3*x-6*x^4)^2 + 72*x^5) / ((1-3*x-6*x^4)^2 - 72*x^5)^2.

%F a(n) = 12*a(n-1) - 54*a(n-2) + 108*a(n-3) - 57*a(n-4) - 72*a(n-5) - 216*a(n-6) + 648*a(n-7) - 216*a(n-8) - 432*a(n-9) - 1944*a(n-10) + 864*a(n-12) + 2592*a(n-13) - 1296*a(n-16).

%o (PARI) my(A=2, B=3, C=4*A*B^2, N=2, 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, A391757.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Dec 18 2025