A392043 - OEIS

%I #18 Dec 31 2025 08:49:10

%S 1,0,0,4,0,0,12,0,0,32,96,0,80,960,0,192,5760,0,448,26880,4032,1024,

%T 107520,64512,2304,387072,580608,5120,1290240,3870720,149504,4055040,

%U 21288960,3065856,12165120,102187008,36548608,35143680,442810368,316407808,102715392

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

%H Seiichi Manyama, <a href="/A392043/b392043.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_20">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,12,0,0,-60,0,0,160,48,0,-240,-288,0,192,576,0,-64,-384,-576).

%F G.f.: (1-2*x^3) * ((1-2*x^3)^3 + 48*x^10) / ((1-2*x^3)^3 - 24*x^10)^2.

%F a(n) = 12*a(n-3) - 60*a(n-6) + 160*a(n-9) + 48*a(n-10) - 240*a(n-12) - 288*a(n-13) + 192*a(n-15) + 576*a(n-16) - 64*a(n-18) - 384*a(n-19) - 576*a(n-20).

%t CoefficientList[Series[(1-2*x^3)*((1-2*x^3)^3+48*x^10)/((1-2*x^3)^3-24*x^10)^2,{x,0,50}],x] (* _Vincenzo Librandi_, Dec 31 2025 *)

%o (PARI) a178618(n, k) = sum(j=0, k, (-1)^(k-j)*binomial(n+1, k-j)*binomial(n+3*j, 3*j));

%o my(A=2, B=3, C=A^3*B, N=2, M=50, x='x+O('x^M), X=1-A*x^3, Y=10); Vec(sum(k=0, (2*N)\3, C^k*a178618(N-1, k)*X^(2*N-3*k)*x^(Y*k))/(X^3-C*x^Y)^N)

%o (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1-2*x^3) * ((1-2*x^3)^3 + 48*x^10) / ((1-2*x^3)^3 - 24*x^10)^2); // _Vincenzo Librandi_, Dec 31 2025

%Y Cf. A178618, A392042.

%K nonn

%O 0,4

%A _Seiichi Manyama_, Dec 29 2025