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G.f. satisfies A(x) = 1 + x*A(x)^6 / (1 + x*A(x)^6).
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%I #20 Aug 26 2023 18:17:21

%S 1,1,5,34,265,2232,19766,181300,1706737,16392049,159959240,1581278838,

%T 15800619070,159321921844,1618981274136,16562211506496,

%U 170426473666497,1762771226922775,18316562635133813,191104193378725552,2001224271292820200

%N G.f. satisfies A(x) = 1 + x*A(x)^6 / (1 + x*A(x)^6).

%C Conjecture: Is a(n)>0 correct? It is correct up to the first 10000 terms.

%H Seiichi Manyama, <a href="/A365218/b365218.txt">Table of n, a(n) for n = 0..956</a>

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

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

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

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

%Y Cf. A291534, A364864, A364865, A364866.

%Y Cf. A002296.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Aug 26 2023