A378610 - OEIS

%I #14 Dec 03 2024 04:23:00

%S 1,4,30,276,2825,30884,353108,4170500,50485764,623084056,7810707894,

%T 99175174284,1272856327470,16486135484248,215212582153840,

%U 2828658852385572,37401956484705132,497174193516767600,6640063367021736728,89058042321373540912,1199031374607501831273

%N Expansion of (1/x) * Series_Reversion( x * (1 - x/(1 - x))^4 ).

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

%F G.f.: exp( Sum_{k>=1} A378613(k) * x^k/k ).

%F a(n) = (1/(n+1)) * [x^n] 1/(1 - x/(1 - x))^(4*(n+1)).

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

%F G.f.: B(x)^4 where B(x) is the g.f. of A243667.

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

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x/(1-x))^4)/x)

%o (PARI) a(n, s=1, t=4, u=-4) = sum(k=0, n\s, binomial(t*(n+1)+k-1, k)*binomial((t+u+1)*(n+1)-(s-1)*k-2, n-s*k))/(n+1);

%Y Cf. A001003, A211789, A369012.

%Y Cf. A243667, A371486, A378613.

%K nonn,new

%O 0,2

%A _Seiichi Manyama_, Dec 01 2024