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A386566 a(n) = Sum_{k=0..n-1} binomial(5*k-1,k) * binomial(5*n-5*k,n-k-1).

%I A386566 #26 Jul 30 2025 05:21:42
%S A386566 0,1,14,181,2284,28506,353630,4370584,53882392,663116347,8150224204,
%T A386566 100073884670,1227826127020,15055154471696,184508186225552,
%U A386566 2260299193652496,27679951219660080,338872887728053465,4147618793911034330,50753529798492061819,620942367878256638264
%N A386566 a(n) = Sum_{k=0..n-1} binomial(5*k-1,k) * binomial(5*n-5*k,n-k-1).
%F A386566 G.f.: g*(g-1)/(5-4*g)^2 where g=1+x*g^5.
%F A386566 G.f.: g/(1-5*g)^2 where g*(1-g)^4 = x.
%F A386566 L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/4) * log( Sum_{k>=0} binomial(5*k-1,k)*x^k ).
%F A386566 a(n) = Sum_{k=0..n-1} binomial(5*k-1+l,k) * binomial(5*n-5*k-l,n-k-1) for every real number l.
%F A386566 a(n) = Sum_{k=0..n-1} 4^(n-k-1) * binomial(5*n,k).
%F A386566 a(n) = Sum_{k=0..n-1} 5^(n-k-1) * binomial(4*n+k,k).
%F A386566 Conjecture D-finite with recurrence 196608*n*(4*n-3)*(2*n-1)*(18270873280*n -32560150837) *(4*n-1)*a(n) +1280*(-1399185802400000*n^5 +1022280893000000*n^4 +17669158913120000*n^3 -48968110172924750*n^2 +49502057719349955*n -17877514345852392)*a(n-1) +125000*(-61298198200000*n^5 +1447969779032500*n^4 -7721498995066250*n^3 +17474948768595875*n^2 -18352567310653770*n +7399184154389181)*a(n-2) +48828125*(5*n-11) *(5*n-14)*(4958243695*n -6717884799) *(5*n-13)*(5*n-12)*a(n-3)=0. - _R. J. Mathar_, Jul 30 2025
%e A386566 (1/4) * log( Sum_{k>=0} binomial(5*k-1,k)*x^k ) = x + 7*x^2 + 181*x^3/3 + 571*x^4 + 28506*x^5/5 + ...
%o A386566 (PARI) a(n) = sum(k=0, n-1, binomial(5*k-1, k)*binomial(5*n-5*k, n-k-1));
%o A386566 (PARI) my(N=30, x='x+O('x^N), g=sum(k=0, N, binomial(5*k, k)/(4*k+1)*x^k)); concat(0, Vec(g*(g-1)/(5-4*g)^2))
%Y A386566 Cf. A000346, A062236, A386565, A386567.
%Y A386566 Cf. A079678, A386367, A386613, A386614.
%Y A386566 Cf. A002294, A118971.
%K A386566 nonn
%O A386566 0,3
%A A386566 _Seiichi Manyama_, Jul 26 2025