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
Keywords
Programs
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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
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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);
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