A369012 Expansion of (1/x) * Series_Reversion( x * (1-x/(1-x))^3 ).
1, 3, 18, 133, 1095, 9636, 88718, 843993, 8230671, 81841987, 826641816, 8457710604, 87472494564, 912995025912, 9604763388534, 101736967518497, 1084125909550959, 11614159795566489, 125011746270524690, 1351312626871871661, 14662950224977228047
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..941
- Index entries for reversions of series
Programs
-
PARI
my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x/(1-x))^3)/x) -
PARI
a(n, s=1, t=3, u=-3) = 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(n-1,n-k).
D-finite with recurrence 96*(3*n+2)*(3*n+1)*(n+1)*a(n) +4*(-4121*n^3 +1922*n^2 -1273*n+124)*a(n-1) +4*(20588*n^3 -76648*n^2 +98677*n -43586)*a(n-2) +(-90073*n^3 +671565*n^2 -1665278*n +1375320)*a(n-3) +210*(n-4)*(3*n-7) *(3*n-8)*a(n-4)=0. - R. J. Mathar, Jan 25 2024
From Seiichi Manyama, Dec 02 2024: (Start)
G.f.: exp( Sum_{k>=1} A378612(k) * x^k/k ).
a(n) = (1/(n+1)) * [x^n] 1/(1 - x/(1 - x))^(3*(n+1)).
G.f.: B(x)^3 where B(x) is the g.f. of A243659.
a(n) = 3 * Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(n,k) * binomial(3*n+k+3,n)/(3*n+k+3). (End)