A322727 G.f.: A(x) = Sum_{n>=0} ((1+x)^n - x*A(x))^n / 2^(n+1).

1, 2, 25, 526, 15554, 590575, 27338911, 1492079652, 93776123520, 6669479907548, 529549414923174, 46432923079132507, 4456442206987048919, 464691980829431905959, 52313905571976063712791, 6323967971227649063422416, 817029035199692037187367976, 112348605647113658601511442450, 16383091093205410144470895801204, 2525293749572595915714422633867944

Offset

0,2

Links

Table of n, a(n) for n=0..19.

Formula

G.f. A(x) satisfies:

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

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

Example

G.f.: A(x) = 1 + 2*x + 25*x^2 + 526*x^3 + 15554*x^4 + 590575*x^5 + 27338911*x^6 + 1492079652*x^7 + 93776123520*x^8 + 6669479907548*x^9 + ...
such that
A(x) = 1/2 + ((1+x) - x*A(x))/2^2 + ((1+x)^2 - x*A(x))^2/2^3 + ((1+x)^3 - x*A(x))^3/2^4 + ((1+x)^4 - x*A(x))^4/2^5 + ((1+x)^5 - x*A(x))^5/2^6 + ...
also,
A(x) = 1/(2 + x*A(x)) + (1+x)/(2 + x*A(x))^2 + (1+x)^4/(2 + x*A(x))^3 + (1+x)^9/(2 + x*A(x))^4 + (1+x)^16/(2 + x*A(x))^5 + (1+x)^25/(2 + x*A(x))^6 + ...

Mathematica

m = 20; A[_] = 0;
Do[A[x_] = Sum[((1+x)^n - x A[x])^n/2^(n+1), {n, 0, k^2}] + O[x]^k, {k, 1, m}];
CoefficientList[A[x], x] // Round (* Jean-François Alcover, Oct 01 2019 *)

Crossrefs

Cf. A322728.

Sequence in context: A074209 A209467 A121252 * A090733 A330767 A197084

Adjacent sequences: A322724 A322725 A322726 * A322728 A322729 A322730

Keyword

nonn

Author

Paul D. Hanna, Jan 31 2019

Status

approved