A369265 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 / (1+x^3) ).
1, 2, 7, 31, 153, 806, 4439, 25250, 147193, 874732, 5279635, 32276245, 199439761, 1243633652, 7815804351, 49455190791, 314807497953, 2014530780524, 12952334769203, 83628832755779, 542022781854953, 3525150296312984, 22998642171764363, 150478455899387966
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Keywords
Links
Programs
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Maple
A369265 := proc(n) add(binomial(n+1,k) * binomial(3*n-3*k+1,n-3*k),k=0..floor(n/3)) ; %/(n+1) ; end proc; seq(A369265(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)^2/(1+x^3))/x) -
PARI
a(n, s=3, t=1, u=2) = sum(k=0, n\s, binomial(t*(n+1), k)*binomial((u+1)*(n+1)-s*k-2, n-s*k))/(n+1);
Formula
a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(n+1,k) * binomial(3*n-3*k+1,n-3*k).
D-finite with recurrence 16*(n+1)*(2*n+1)*a(n) +4*(-89*n^2+15*n+2)*a(n-1) +3*(345*n^2-603*n+274)*a(n-2) +18*(-41*n^2+45*n+94)*a(n-3) +54*(-4*n^2+57*n-137)*a(n-4) +486*(n-4)*(n-5)*a(n-5) -243*(n-4)*(n-5)*a(n-6)=0. - R. J. Mathar, Jan 25 2024
a(n) = (1/(n+1)) * [x^n] ( 1/(1-x)^2 * (1+x^3) )^(n+1). - Seiichi Manyama, Feb 14 2024