A369114 Expansion of (1/x) * Series_Reversion( x * ((1-x)^3-x^3) ).

1, 3, 15, 92, 630, 4620, 35494, 282015, 2298417, 19108265, 161418543, 1381606044, 11955789440, 104427062460, 919430773992, 8151530382264, 72711166411422, 652075100808960, 5875868463764446, 53175058170610530, 483082193418731280, 4404057834071995110

Offset

0,2

Links

Table of n, a(n) for n=0..21.

Index entries for reversions of series

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(n+k,k) * binomial(4*n+2,n-3*k).

D-finite with recurrence 81*n*(n-1)*(n+1)*a(n) -945*n^2*(n-1)*a(n-1) +441*(n-1)*(3*n^2+9*n-20)*a(n-2) +3*(1039*n^3 -12393*n^2 +37406*n-33232)*a(n-3) -448*(2*n-5) *(4*n-13)*(4*n-11)*a(n-4)=0. - R. J. Mathar, Jan 25 2024

Maple

A369114 := proc(n)
    add(binomial(n+k, k) * binomial(4*n+2, n-3*k), k=0..floor(n/3)) ;
    %/(n+1) ;
end proc;
seq(A369114(n), n=0..70) ; # R. J. Mathar, Jan 25 2024

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x*((1-x)^3-x^3))/x)
(PARI) a(n) = sum(k=0, n\3, binomial(n+k, k)*binomial(4*n+2, n-3*k))/(n+1);

Crossrefs

Cf. A368011, A369102.

Cf. A097188.

Sequence in context: A124553 A369623 A366053 * A020108 A323696 A231657

Adjacent sequences: A369111 A369112 A369113 * A369115 A369116 A369117

Keyword

nonn

Author

Seiichi Manyama, Jan 13 2024

Status

approved