cp's OEIS Frontend

A386367 a(n) = Sum_{k=0..n-1} binomial(5*k,k) * binomial(5*n-5*k-2,n-k-1).

%I A386367 #34 Jul 29 2025 08:39:50
%S A386367 0,1,13,163,2021,24930,306655,3765448,46182101,565939603,6931070490,
%T A386367 84845250370,1038235255415,12700966517968,155336699256808,
%U A386367 1899439862390640,23222289820948405,283872591297526505,3469680960837171415,42404345427419774621,518193229118757697930
%N A386367 a(n) = Sum_{k=0..n-1} binomial(5*k,k) * binomial(5*n-5*k-2,n-k-1).
%F A386367 G.f.: g*(1-g)/(1-5*g)^2 where g*(1-g)^4 = x.
%F A386367 L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/5) * log( Sum_{k>=0} binomial(5*k,k)*x^k ).
%F A386367 G.f.: (g-1)/(5-4*g)^2 where g=1+x*g^5.
%F A386367 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.
%F A386367 a(n) = Sum_{k=0..n-1} 4^(n-k-1) * binomial(5*n-1,k).
%F A386367 a(n) = Sum_{k=0..n-1} 5^(n-k-1) * binomial(4*n+k-1,k).
%e A386367 (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 + ...
%o A386367 (PARI) a(n) = sum(k=0, n-1, binomial(5*k, k)*binomial(5*n-5*k-2, n-k-1));
%o A386367 (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))
%Y A386367 Cf. A000302, A036829, A308523, A386368.
%Y A386367 Cf. A079678, A386566, A386613, A386614.
%Y A386367 Cf. A002294, A118971.
%K A386367 nonn
%O A386367 0,3
%A A386367 _Seiichi Manyama_, Jul 19 2025