%I #15 Jul 13 2023 07:51:01
%S 1,2,22,350,6538,133658,2895214,65294502,1516963346,36056007602,
%T 872615973766,21430572885422,532737957899290,13379121740808266,
%U 338941379999841758,8651415618928816886,222278432539991439906,5743974149517874477922,149192980850883703986166
%N Expansion of g.f. A(x) satisfying A(x) = 1 + x*(A(x)^4 + A(x)^7).
%H Seiichi Manyama, <a href="/A363304/b363304.txt">Table of n, a(n) for n = 0..500</a>
%F G.f.: A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
%F (1) A(x) = 1 + x*(A(x)^4 + A(x)^7).
%F (2) a(n) = Sum_{k=0..n} binomial(n, k)*binomial(4*n+3*k+1, n)/(4*n+3*k+1) for n >= 0.
%e G.f.: A(x) = 1 + 2*x + 22*x^2 + 350*x^3 + 6538*x^4 + 133658*x^5 + 2895214*x^6 + 65294502*x^7 + 1516963346*x^8 + 36056007602*x^9 + ...
%e where A(x) = 1 + x*(A(x)^4 + A(x)^7).
%e RELATED SERIES.
%e A(x)^4 = 1 + 8*x + 112*x^2 + 1960*x^3 + 38528*x^4 + 813064*x^5 + 17998512*x^6 + 412364968*x^7 + ...
%e A(x)^7 = 1 + 14*x + 238*x^2 + 4578*x^3 + 95130*x^4 + 2082150*x^5 + 47295990*x^6 + 1104598378*x^7 + ...
%o (PARI) {a(n) = sum(k=0, n, binomial(n, k)*binomial(4*n+3*k+1, n)/(4*n+3*k+1) )}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A027307, A363311, A363111.
%K nonn
%O 0,2
%A _Paul D. Hanna_, May 29 2023