A381910 Expansion of (1/x) * Series_Reversion( x / ((1+x)^3 * B(x)) ), where B(x) is the g.f. of A002293.
1, 4, 26, 222, 2243, 25243, 305217, 3878731, 51097713, 691596081, 9558970897, 134347855874, 1914131985782, 27582542400252, 401284140631911, 5886072268606617, 86951528919335670, 1292467847124221832, 19316795168721092789, 290107272994659617741, 4375905051887803660504
Offset: 0
Keywords
Programs
-
PARI
a(n) = sum(k=0, n, binomial(n+4*k+1, k)*binomial(3*n+3, n-k)/(n+4*k+1));
Formula
G.f. A(x) satisfies A(x) = (1 + x*A(x))^3 * B(x*A(x)).
a(n) = Sum_{k=0..n} binomial(n+4*k+1,k) * binomial(3*n+3,n-k)/(n+4*k+1).
a(n) = binomial(3*(1 + n), n)*hypergeom([(1+n)/4, (2+n)/4, (3+n)/4, (4+n)/4, -n], [(2+n)/3, (3+n)/3, (4+n)/3, 4+2*n], -2^8/3^3)/(1 + n). - Stefano Spezia, Mar 10 2025