A302105 G.f. A(x) satisfies: A(x) = Sum_{n>=0} (4 + x*A(x)^n)^n / 5^(n+1).

1, 1, 10, 175, 3835, 95090, 2551480, 72360700, 2139052845, 65329175385, 2049247480265, 65752776679275, 2151923601749290, 71691421965972905, 2428004656549037580, 83523871228996755395, 2917260885363111908770, 103451501815230690971935, 3726040763307222530311125, 136400452641372633368206185, 5080478361492407723101242440

Offset: 0

Author

Paul D. Hanna, Apr 05 2018

Keywords

• nonn

Comments

Compare to: G(x) = Sum_{n>=0} (4 + x*G(x)^k)^n / 5^(n+1) holds when G(x) = 1 + x*G(x)^(k+1) for fixed k.

Examples

			G.f.: A(x) = 1 + x + 10*x^2 + 175*x^3 + 3835*x^4 + 95090*x^5 + 2551480*x^6 + 72360700*x^7 + 2139052845*x^8 + 65329175385*x^9 + 2049247480265*x^10 + ...
such that
A(x) = 4/5 + (4 + x*A(x))/5^2 + (4 + x*A(x)^2)^2/5^3 + (4 + x*A(x)^3)^3/5^4 + (4 + x*A(x)^4)^4/5^5 + (4 + x*A(x)^5)^5/5^6 + (4 + x*A(x)^6)^6/5^7 + ...
Also, due to a series identity,
A(x) = 1 + x*A(x)/(5 - 4*A(x))^2 + x^2*A(x)^4/(5 - 4*A(x)^2)^3 + x^3*A(x)^9/(5 - 4*A(x)^3)^4 + x^4*A(x)^16/(5 - 4*A(x)^4)^5 + x^5*A(x)^25/(5 - 4*A(x)^5)^6 + x^6*A(x)^36/(5 - 4*A(x)^6)^7 + ... + x^n * A(x)^(n^2) / (5 - 4*A(x)^n)^(n+1) + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A300050, A302103, A302104.

Programs

PARI

{a(n) = my(A=1); for(i=0, n, A = sum(m=0, n, x^m * A^(m^2) / (5 - 4*A^m + x*O(x^n))^(m+1) )); polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies:

(1) A(x) = Sum_{n>=0} (4 + x*A(x)^n)^n / 5^(n+1).

(2) A(x) = Sum_{n>=0} x^n * A(x)^(n^2) / (5 - 4*A(x)^n)^(n+1).