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A369012 Expansion of (1/x) * Series_Reversion( x * (1-x/(1-x))^3 ).

%I A369012 #28 Dec 03 2024 04:22:26
%S A369012 1,3,18,133,1095,9636,88718,843993,8230671,81841987,826641816,
%T A369012 8457710604,87472494564,912995025912,9604763388534,101736967518497,
%U A369012 1084125909550959,11614159795566489,125011746270524690,1351312626871871661,14662950224977228047
%N A369012 Expansion of (1/x) * Series_Reversion( x * (1-x/(1-x))^3 ).
%H A369012 Seiichi Manyama, <a href="/A369012/b369012.txt">Table of n, a(n) for n = 0..941</a>
%H A369012 <a href="/index/Res#revert">Index entries for reversions of series</a>
%F A369012 a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(3*n+k+2,k) * binomial(n-1,n-k).
%F A369012 D-finite with recurrence 96*(3*n+2)*(3*n+1)*(n+1)*a(n) +4*(-4121*n^3 +1922*n^2 -1273*n+124)*a(n-1) +4*(20588*n^3 -76648*n^2 +98677*n -43586)*a(n-2) +(-90073*n^3 +671565*n^2 -1665278*n +1375320)*a(n-3) +210*(n-4)*(3*n-7) *(3*n-8)*a(n-4)=0. - _R. J. Mathar_, Jan 25 2024
%F A369012 From _Seiichi Manyama_, Dec 02 2024: (Start)
%F A369012 G.f.: exp( Sum_{k>=1} A378612(k) * x^k/k ).
%F A369012 a(n) = (1/(n+1)) * [x^n] 1/(1 - x/(1 - x))^(3*(n+1)).
%F A369012 G.f.: B(x)^3 where B(x) is the g.f. of A243659.
%F A369012 a(n) = 3 * Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(n,k) * binomial(3*n+k+3,n)/(3*n+k+3). (End)
%o A369012 (PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x/(1-x))^3)/x)
%o A369012 (PARI) a(n, s=1, t=3, u=-3) = 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 A369012 Cf. A369013, A369014.
%Y A369012 Cf. A243659, A378612.
%Y A369012 Cf. A001003, A211789, A378610.
%K A369012 nonn
%O A369012 0,2
%A A369012 _Seiichi Manyama_, Jan 11 2024