OFFSET
0,3
FORMULA
G.f.: g*(1-g)/(1-5*g)^2 where g*(1-g)^4 = x.
L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/5) * log( Sum_{k>=0} binomial(5*k,k)*x^k ).
G.f.: (g-1)/(5-4*g)^2 where g=1+x*g^5.
a(n) = Sum_{k=0..n-1} binomial(5*k-2+l,k) * binomial(5*n-5*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 4^(n-k-1) * binomial(5*n-1,k).
a(n) = Sum_{k=0..n-1} 5^(n-k-1) * binomial(4*n+k-1,k).
EXAMPLE
(1/5) * log( Sum_{k>=0} binomial(5*k,k)*x^k ) = x + 13*x^2/2 + 163*x^3/3 + 2021*x^4/4 + 4986*x^5 + ...
PROG
(PARI) a(n) = sum(k=0, n-1, binomial(5*k, k)*binomial(5*n-5*k-2, n-k-1));
(PARI) my(N=30, x='x+O('x^N), g=x*sum(k=0, N, binomial(5*k+3, k)/(k+1)*x^k)); concat(0, Vec(g*(1-g)/(1-5*g)^2))
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Jul 19 2025
STATUS
approved
