A386770 - OEIS

%I #13 Aug 03 2025 11:18:43

%S 1,13,244,5397,130961,3372268,90497184,2503434117,70883043571,

%T 2044268649573,59842331451024,1773506049794412,53107658756034156,

%U 1604418047921589928,48841208603255888264,1496711470907670605157,46134317696761847385591,1429405788411234205692583

%N Expansion of (1/x) * Series_Reversion( x * (1-2*x)^2 / (1+3*x)^3 ).

%H <a href="/index/Res#revert">Index entries for reversions of series</a>

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

%F a(n) = (1/(n+1)) * [x^n] ( (1+3*x)^3 / (1-2*x)^2 )^(n+1).

%F D-finite with recurrence 32*(n+1)*(2*n+1)*a(n) +48*(-81*n^2+27*n-7)*a(n-1) +162*(414*n^2-891*n+605)*a(n-2) -32805*(3*n-4)*(3*n-5)*a(n-3)=0. - _R. J. Mathar_, Aug 03 2025

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serreverse(x*(1-2*x)^2/(1+3*x)^3)/x)

%o (PARI) a(n) = sum(k=0, n, 3^k*2^(n-k)*binomial(3*(n+1), k)*binomial(3*n-k+1, n-k))/(n+1);

%Y Cf. A385474, A386764.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Aug 02 2025