A395669 - OEIS

%I #17 May 04 2026 10:50:00

%S 1,16,268,4572,78746,1364104,23719728,413545356,7224055351,

%T 126380687416,2213511996356,38804671881288,680789488507506,

%U 11951216432486432,209912513875637888,3688563381314685804,64839772922717680659,1140167947961926829368,20054925618261969731104

%N Expansion of g^7*(4-3*g)/(7-6*g)^2, where g = 1+x*g^7 is the g.f. of A002296.

%F Sum_{k>=1} a(k-1) * x^k/k = (1/14) * log( Sum_{k>=0} binomial(7*k+7,k) * x^k ).

%F a(n) = A395667(n)/(n+1) = (1/14) * (binomial(7*n+6,n) + Sum_{k=0..n+1} 6^(n+1-k) * binomial(7*n+7,k)).

%F a(n) = (1/12) * Sum_{k=0..n+1} 6^(n+1-k) * binomial(7*n+6,k).

%F a(n) = (1/(n+1)) * Sum_{k=0..n} 6^k * binomial(k+2,2) * binomial(7*n+7,n-k).

%o (PARI) a(n) = (binomial(7*n+6, n)+sum(k=0, n+1, 6^(n+1-k)*binomial(7*n+7, k)))/14;

%Y Cf. A395655, A395668.

%Y Cf. A002296, A004369, A395667.

%K nonn

%O 0,2

%A _Seiichi Manyama_, May 03 2026