A304860 G.f. A(x) satisfies: x = Sum_{n>=0} ( (1+x)^(n^2) - A(x)^n ) / 2^(n+1).

1, 2, 32, 608, 17750, 683504, 32183336, 1782735248, 113381031512, 8138225237204, 650735042088080, 57369033007665680, 5529284312514428840, 578479328396134930928, 65297339893598788494368, 7910610591246432715704704, 1023854667471171305890388408, 141001918216059025744295715872, 20587944237516075824024078357264, 3176963079503660078673757802123360

Offset: 0

Author

Paul D. Hanna, May 28 2018

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + 2*x + 32*x^2 + 608*x^3 + 17750*x^4 + 683504*x^5 + 32183336*x^6 + 1782735248*x^7 + 113381031512*x^8 + 8138225237204*x^9 + ...
such that
x = ((1+x) - A(x))/2^2 + ((1+x)^4 - A(x)^2)/2^3 + ((1+x)^9 - A(x)^3)/2^4 + ((1+x)^16 - A(x)^4)/2^5 + ((1+x)^25 - A(x)^5)/2^6 + ((1+x)^36 - A(x)^6)/2^7 + ...
RELATED SERIES.
G(x) = Sum_{n>=0} (1+x)^(n^2) / 2^(n+1) = 1 + 3*x + 36*x^2 + 744*x^3 + 21606*x^4 + 807912*x^5 + 36948912*x^6 + 1997801520*x^7 + 124666314300*x^8 + ... + A173217(n)*x^n + ...
1/(2 - A(x)) = G(x) - x = 1 + 2*x + 36*x^2 + 744*x^3 + 21606*x^4 + 807912*x^5 + 36948912*x^6 + 1997801520*x^7 + 124666314300*x^8 + ...
Let F(x) satisfy
x = Sum_{n>=0} ( F(x)^n - A(x)^n ) / 2^(n+1), then
F(x) = 1 + 3*x + 27*x^2 + 555*x^3 + 16737*x^4 + 652815*x^5 + 30967917*x^6 + 1724292411*x^7 + 110091861729*x^8 + 7926482395935*x^9 + ...
where 1/(2 - F(x)) = x + 1/(2 - A(x)).
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..60

Crossrefs

Cf. A173217.

Formula

G.f. A(x) satisfies:

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

(2) A(x) = 2 - 1/(G(x) - x), where G(x) = Sum_{n>=0} (1+x)^(n^2) / 2^(n+1) is the g.f. of A173217.