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A386567 a(n) = Sum_{k=0..n-1} binomial(6*k-1,k) * binomial(6*n-6*k,n-k-1).

%I A386567 #28 Jul 29 2025 08:39:35
%S A386567 0,1,17,268,4129,62955,954392,14417376,217279857,3269099590,
%T A386567 49125066135,737516631908,11064270530632,165889863957065,
%U A386567 2486052264852180,37241727274394640,557707191712371729,8349517132932620730,124971965902300790390,1870139909398530770760
%N A386567 a(n) = Sum_{k=0..n-1} binomial(6*k-1,k) * binomial(6*n-6*k,n-k-1).
%F A386567 G.f.: g*(g-1)/(6-5*g)^2 where g=1+x*g^6.
%F A386567 G.f.: g/(1-6*g)^2 where g*(1-g)^5 = x.
%F A386567 L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/5) * log( Sum_{k>=0} binomial(6*k-1,k)*x^k ).
%F A386567 a(n) = Sum_{k=0..n-1} binomial(6*k-1+l,k) * binomial(6*n-6*k-l,n-k-1) for every real number l.
%F A386567 a(n) = Sum_{k=0..n-1} 5^(n-k-1) * binomial(6*n,k).
%F A386567 a(n) = Sum_{k=0..n-1} 6^(n-k-1) * binomial(5*n+k,k).
%e A386567 (1/5) * log( Sum_{k>=0} binomial(6*k-1,k)*x^k ) = x + 17*x^2/2 + 268*x^3/3 + 4129*x^4/4 + 12591*x^5 + ...
%o A386567 (PARI) a(n) = sum(k=0, n-1, binomial(6*k-1, k)*binomial(6*n-6*k, n-k-1));
%o A386567 (PARI) my(N=20, x='x+O('x^N), g=sum(k=0, N, binomial(6*k, k)/(5*k+1)*x^k)); concat(0, Vec(g*(g-1)/(6-5*g)^2))
%Y A386567 Cf. A000346, A062236, A386565, A386566.
%Y A386567 Cf. A079679, A386368, A386615, A386616.
%Y A386567 Cf. A002295, A130564.
%K A386567 nonn
%O A386567 0,3
%A A386567 _Seiichi Manyama_, Jul 26 2025