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A386368 a(n) = Sum_{k=0..n-1} binomial(6*k,k) * binomial(6*n-6*k-2,n-k-1).

%I A386368 #35 Jul 30 2025 04:42:10
%S A386368 0,1,16,246,3736,56421,849432,12763878,191548464,2871970110,
%T A386368 43031833656,644432826478,9646983339456,144366433138955,
%U A386368 2159869510669320,32306874783230556,483151884326658144,7224464127509984490,108011596038055519680,1614676987907480393940
%N A386368 a(n) = Sum_{k=0..n-1} binomial(6*k,k) * binomial(6*n-6*k-2,n-k-1).
%F A386368 G.f.: g*(1-g)/(1-6*g)^2 where g*(1-g)^5 = x.
%F A386368 L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/6) * log( Sum_{k>=0} binomial(6*k,k)*x^k ).
%F A386368 G.f.: (g-1)/(6-5*g)^2 where g=1+x*g^6.
%F A386368 a(n) = Sum_{k=0..n-1} binomial(6*k-2+l,k) * binomial(6*n-6*k-l,n-k-1) for every real number l.
%F A386368 a(n) = Sum_{k=0..n-1} 5^(n-k-1) * binomial(6*n-1,k).
%F A386368 a(n) = Sum_{k=0..n-1} 6^(n-k-1) * binomial(5*n+k-1,k).
%F A386368 Conjecture D-finite with recurrence 48828125*(n-1)*(5*n-4)*(5*n-3) *(432862082629612805*n -769306661967834399) *(5*n-2)*(5*n-1)*a(n) +1125000*(-405245406115816219575000*n^6 +2613180799468910510392500*n^5 -7667164406968651479521250*n^4 +13834502135358262506660375*n^3 -16251583347734702117341345*n^2 +11251247074043948959380314*n -3395699069351241765495720)*a(n-1) +33592320*(142281690918326440537500*n^6 -1266424338521609272012500*n^5 +5236041263583271687953750*n^4 -12786608152035075786775875*n^3 +18838556229131595646260055*n^2 -15323925851720394901667853*n +5240681406952416812161236)*a(n-2) -53444359913472*(6*n-17) *(395547729523405*n -538181211711288)*(6*n-13) *(3*n-7)*(2*n-5) *(3*n-8)*a(n-3)=0. - _R. J. Mathar_, Jul 30 2025
%e A386368 (1/6) * log( Sum_{k>=0} binomial(6*k,k)*x^k ) = x + 8*x^2 + 82*x^3 + 934*x^4 + 56421*x^5/5 + ...
%p A386368 A386368 := proc(n::integer)
%p A386368     add(binomial(6*k,k)*binomial(6*n-6*k-2,n-k-1),k=0..n-1) ;
%p A386368 end proc:
%p A386368 seq(A386368(n),n=0..80) ; # _R. J. Mathar_, Jul 30 2025
%o A386368 (PARI) a(n) = sum(k=0, n-1, binomial(6*k, k)*binomial(6*n-6*k-2, n-k-1));
%o A386368 (PARI) my(N=20, x='x+O('x^N), g=x*sum(k=0, N, binomial(6*k+4, k)/(k+1)*x^k)); concat(0, Vec(g*(1-g)/(1-6*g)^2))
%Y A386368 Cf. A000302, A036829, A308523, A386367.
%Y A386368 Cf. A079679, A386567, A386615, A386616.
%Y A386368 Cf. A002295, A130564.
%K A386368 nonn
%O A386368 0,3
%A A386368 _Seiichi Manyama_, Jul 19 2025