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A363304 Expansion of g.f. A(x) satisfying A(x) = 1 + x*(A(x)^4 + A(x)^7).

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