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

1, 5, 43, 451, 5253, 65297, 848503, 11387047, 156602761, 2195519965, 31261365155, 450840279787, 6571775541069, 96669928040745, 1433170971310191, 21392403565317839, 321228841377255953, 4849129915768191413, 73545708989920501147, 1120169585882592246419

Offset: 0

Author

Seiichi Manyama, Jan 12 2024

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..825
• Index entries for reversions of series

Crossrefs

Cf. A344553, A369024.

Programs

Maple

A369023 := proc(n)
    add(binomial(3*n+k+2,k) * binomial(3*n+1,n-k),k=0..n) ;
    %/(n+1) ;
end proc;
seq(A369023(n),n=0..70) ; % R. J. Mathar, Jan 25 2024

PARI

my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-2*x)^3/(1-x))/x)

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);

Formula

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

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