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A363111 Expansion of g.f. A(x) = F(x*F(x)^7), where F(x) = 1 + x*F(x)^4 is the g.f. of A002293.

%I A363111 #9 May 31 2023 09:07:40
%S A363111 1,1,11,127,1547,19652,258069,3481034,47999915,674086924,9612919156,
%T A363111 138878011335,2028718584989,29918897595468,444889269572286,
%U A363111 6663228661354420,100430376524360459,1522215623202615036,23187346871707554564,354783440893854307244
%N A363111 Expansion of g.f. A(x) = F(x*F(x)^7), where F(x) = 1 + x*F(x)^4 is the g.f. of A002293.
%C A363111 Compare the g.f. A(x) = F(x*F(x)^7) to F(-x*F(x)^7) = 1/F(x), where F(x) = 1 + x*F(x)^4 is the g.f. of A002293.
%F A363111 G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined as follows; here, F(x) is the g.f. of A002293.
%F A363111 (1) A(x) = F(x*F(x)^7), where F(x) = 1 + x*F(x)^4.
%F A363111 (2) A(x) = B(x/A(x)^3) where B(x) = A(x*B(x)^3) = F( x*B(x)^3 * F(x*B(x)^3)^7 ).
%F A363111 (3) a(n) = Sum_{k=1..n} 7*k* binomial(4*k+1, k) * binomial(4*n+3*k, n-k) / ((4*k+1)*(4*n+3*k)) for n > 0, with a(0) = 1.
%e A363111 G.f.: A(x) = 1 + x + 11*x^2 + 127*x^3 + 1547*x^4 + 19652*x^5 + 258069*x^6 + 3481034*x^7 + 47999915*x^8 + 674086924*x^9 + ...
%e A363111 such that A(x) = F(x*F(x)^7) where F(x) = 1 + x*F(x)^4 begins
%e A363111 F(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 + 53820*x^7 + ... + A002293(n)*x^n + ...
%e A363111 RELATED SERIES.
%e A363111 Let B(x) = A(x*B(x)^3) = ( Series_Reversion( x/A(x)^3 )/x )^(1/3) which begins
%e A363111 B(x) = 1 + x + 14*x^2 + 238*x^3 + 4578*x^4 + 95130*x^5 + 2082150*x^6 + 47295990*x^7 + 1104598378*x^8 + ...
%e A363111 then
%e A363111 ( (B(x) - 1)/x )^(1/7) = 1 + 2*x + 22*x^2 + 350*x^3 + 6538*x^4 + 133658*x^5 + 2895214*x^6 + 65294502*x^7 + ... + A363304(n)*x^n + ...
%e A363111 is an integer series.
%o A363111 (PARI) {a(n) = if(n==0, 1, sum(k=1, n, 7*k* binomial(4*k+1, k) * binomial(4*n+3*k, n-k) / ((4*k+1)*(4*n+3*k)) ) )}
%o A363111 for(n=0, 30, print1(a(n), ", "))
%o A363111 (PARI) /* G.f. A(x) = F(x*F(x)^7), where F(x) = 1 + x*F(x)^4 */
%o A363111 {a(n) = my(F = 1); for(i=1,n, F = 1 + x*F^4 + x*O(x^n));
%o A363111 polcoeff( subst(F, x, x*F^7), n)}
%o A363111 for(n=0, 30, print1(a(n), ", "))
%Y A363111 Cf. A363304, A363308, A363309, A002293.
%K A363111 nonn
%O A363111 0,3
%A A363111 _Paul D. Hanna_, May 30 2023