A383637 - OEIS

%I #26 May 04 2025 13:59:02

%S 1,3,22,90,511,2373,12412,60420,307021,1520343,7646002,38097150,

%T 190884331,953225913,4769716792,23837822280,119221396441,596010127083,

%U 2980341200782,14900834307810,74506786627351,372526087871853,1862653975153972,9313199268385740,46566208164081061

%N Expansion of 1/((1-x) * (1+3*x) * (1-5*x)).

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,13,-15).

%F a(n) = Sum_{k=0..floor(n/2)} 16^k * binomial(n+2,2*k+2).

%F a(n) = (5^(n+2) + (-3)^(n+2) - 2)/32 = (A120612(n+2) - 1)/16.

%F a(n) = 3*a(n-1) + 13*a(n-2) - 15*a(n-3).

%F a(n) = Sum_{k=0..n} 4^k * (-3)^(n-k) * binomial(n+2,k+2) * Stirling2(k+2,2).

%F a(n) = Sum_{k=0..n} (-4)^k * 5^(n-k) * binomial(n+2,k+2) * Stirling2(k+2,2).

%o (PARI) a(n) = (5^(n+2)+(-3)^(n+2)-2)/32;

%Y Cf. A079773, A120612.

%K nonn,easy

%O 0,2

%A _Seiichi Manyama_, May 03 2025