A300050 G.f. A(x) satisfies: A(x) = Sum_{n>=0} (1 + x*A(x)^n)^n / 2^(n+1).

1, 1, 4, 28, 250, 2558, 28594, 340486, 4255344, 55300776, 742732646, 10267434138, 145692400018, 2118364506746, 31529605958892, 480186833802260, 7483472464151002, 119397596900634238, 1951747376021480874, 32721008993895160926, 563260078608381337148, 9967709437187109520736, 181544883799333028286098, 3406426523158387599683478, 65894531117591548919270114

Offset

0,3

Comments

Compare to: G(x) = Sum_{n>=0} (1 + x*G(x)^k)^n / 2^(n+1) holds when G(x) = 1 + x*G(x)^(k+1) for fixed k.

Links

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

Formula

G.f. A(x) satisfies:

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

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

Example

G.f.: A(x) = 1 + x + 4*x^2 + 28*x^3 + 250*x^4 + 2558*x^5 + 28594*x^6 + 340486*x^7 + 4255344*x^8 + 55300776*x^9 + 742732646*x^10 + ...
such that
A(x) = 1/2 + (1 + x*A(x))/2^2 + (1 + x*A(x)^2)^2/2^3 + (1 + x*A(x)^3)^3/2^4 + (1 + x*A(x)^4)^4/2^5 + (1 + x*A(x)^5)^5/2^6 + (1 + x*A(x)^6)^6/2^7 + ...
Also, due to a series identity,
A(x) = 1 + x*A(x)/(2 - A(x))^2 + x^2*A(x)^4/(2 - A(x)^2)^3 + x^3*A(x)^9/(2 - A(x)^3)^4 + x^4*A(x)^16/(2 - A(x)^4)^5 + x^5*A(x)^25/(2 - A(x)^5)^6 + x^6*A(x)^36/(2 - A(x)^6)^7 + ... + x^n * A(x)^(n^2) / (2 - A(x)^n)^(n+1) + ...

Prog

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

Crossrefs

Cf. A302103, A302104, A302105, A300279.

Sequence in context: A228714 A371693 A230640 * A191801 A365562 A064340

Adjacent sequences: A300047 A300048 A300049 * A300051 A300052 A300053

Keyword

nonn

Author

Paul D. Hanna, Feb 27 2018

Status

approved