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A378610 Expansion of (1/x) * Series_Reversion( x * (1 - x/(1 - x))^4 ).

%I A378610 #14 Dec 03 2024 04:23:00
%S A378610 1,4,30,276,2825,30884,353108,4170500,50485764,623084056,7810707894,
%T A378610 99175174284,1272856327470,16486135484248,215212582153840,
%U A378610 2828658852385572,37401956484705132,497174193516767600,6640063367021736728,89058042321373540912,1199031374607501831273
%N A378610 Expansion of (1/x) * Series_Reversion( x * (1 - x/(1 - x))^4 ).
%H A378610 <a href="/index/Res#revert">Index entries for reversions of series</a>
%F A378610 G.f.: exp( Sum_{k>=1} A378613(k) * x^k/k ).
%F A378610 a(n) = (1/(n+1)) * [x^n] 1/(1 - x/(1 - x))^(4*(n+1)).
%F A378610 a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(4*n+k+3,k) * binomial(n-1,n-k).
%F A378610 G.f.: B(x)^4 where B(x) is the g.f. of A243667.
%F A378610 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 A378610 (PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x/(1-x))^4)/x)
%o A378610 (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 A378610 Cf. A001003, A211789, A369012.
%Y A378610 Cf. A243667, A371486, A378613.
%K A378610 nonn
%O A378610 0,2
%A A378610 _Seiichi Manyama_, Dec 01 2024