A369617 Expansion of (1/x) * Series_Reversion( x / (1/(1-x)^3 + x) ).

1, 4, 22, 146, 1079, 8525, 70468, 601816, 5268241, 47019566, 426250277, 3914020148, 36328457669, 340278596273, 3212416054283, 30534649412247, 291981031204917, 2806832429353512, 27109863184695640, 262951127248539898, 2560229132085602215

Offset: 0

Author

Seiichi Manyama, Jan 27 2024

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..984
• Index entries for reversions of series

Crossrefs

Cf. A007317, A369616, A370844.

Cf. A006632, A274735.

Programs

Maple

A369617 := proc(n)
    add(binomial(n+1,k) * binomial(4*n-4*k+2,n-k),k=0..n) ;
    %/(n+1) ;
end proc;
seq(A369617(n),n=0..70) ; # R. J. Mathar, Jan 28 2024

PARI

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

PARI

a(n) = sum(k=0, n, binomial(n+1, k)*binomial(4*n-4*k+2, n-k))/(n+1);

Formula

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

D-finite with recurrence 3*(3*n+2)*(3*n+1)*(n+1)*a(n) +4*(-91*n^3 -32*n^2 +n+2)*a(n-1) +2*(n-1)*(465*n^2 -337*n+86)*a(n-2) -4*(n-1)*(n-2) *(219*n-187)*a(n-3) +283*(n-1)*(n-2)*(n-3)*a(n-4)=0. - R. J. Mathar, Jan 28 2024