cp's OEIS Frontend

A369023 Expansion of (1/x) * Series_Reversion( x * (1-2*x)^3 / (1-x) ).

%I A369023 #20 Jan 25 2024 11:01:08
%S A369023 1,5,43,451,5253,65297,848503,11387047,156602761,2195519965,
%T A369023 31261365155,450840279787,6571775541069,96669928040745,
%U A369023 1433170971310191,21392403565317839,321228841377255953,4849129915768191413,73545708989920501147,1120169585882592246419
%N A369023 Expansion of (1/x) * Series_Reversion( x * (1-2*x)^3 / (1-x) ).
%H A369023 Seiichi Manyama, <a href="/A369023/b369023.txt">Table of n, a(n) for n = 0..825</a>
%H A369023 <a href="/index/Res#revert">Index entries for reversions of series</a>
%F A369023 a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(3*n+k+2,k) * binomial(3*n+1,n-k).
%F A369023 D-finite with recurrence 27*(3*n+2)*(3*n+1)*(n+1)*a(n) +9*(-689*n^3 +263*n^2 -132*n +16)*a(n-1) +6*(6039*n^3 -20979*n^2 +23222*n -8050)*a(n-2) +(43*n^3 -5790*n^2 +25097*n -27570)*a(n-3) -15*(3*n-10)*(3*n-8)*(n-3)*a(n-4)=0. - _R. J. Mathar_, Jan 25 2024
%p A369023 A369023 := proc(n)
%p A369023     add(binomial(3*n+k+2,k) * binomial(3*n+1,n-k),k=0..n) ;
%p A369023     %/(n+1) ;
%p A369023 end proc;
%p A369023 seq(A369023(n),n=0..70) ; % _R. J. Mathar_, Jan 25 2024
%o A369023 (PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-2*x)^3/(1-x))/x)
%o A369023 (PARI) a(n, s=1, t=3, u=-1) = 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 A369023 Cf. A344553, A369024.
%K A369023 nonn
%O A369023 0,2
%A A369023 _Seiichi Manyama_, Jan 12 2024