A301581 G.f.: Sum_{n>=0} 2^n * ((1+x)^n - 1)^n.

1, 2, 16, 232, 4748, 125440, 4058312, 155336672, 6864980968, 343995674280, 19270975801600, 1193481831243584, 80966964261458368, 5971270693661978816, 475655179279901897536, 40699219246551726635840, 3722813577249648564213392, 362519587815189751405383520, 37442485808471509306691295808, 4088344078912544484116541775616, 470550859964811044524886252649760

Offset: 0

Author

Paul D. Hanna, Mar 24 2018

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + 2*x + 16*x^2 + 232*x^3 + 4748*x^4 + 125440*x^5 + 4058312*x^6 + 155336672*x^7 + 6864980968*x^8 + 343995674280*x^9 + ...
such that
A(x) = 1 + 2*((1+x)-1) + 4*((1+x)^2-1)^2 + 8*((1+x)^3-1)^3 + 16*((1+x)^4-1)^4 + 32*((1+x)^5-1)^5 + 64*((1+x)^6-1)^6 + 128*((1+x)^7-1)^7 + ...
Also,
A(x) = 1/3 + 2*(1+x)/(1 + 2*(1+x))^2 + 4*(1+x)^4/(1 + 2*(1+x)^2)^3 + 8*(1+x)^9/(1 + 2*(1+x)^3)^4 + 16*(1+x)^16/(1 + 2*(1+x)^4)^5 + 32*(1+x)^25/(1 + 2*(1+x)^5)^6 + ...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..345

Crossrefs

Cf. A122400, A195005, A301582, A301583, A301463.

Programs

Mathematica

nmax = 20; CoefficientList[Series[1 + Sum[2^j*((1 + x)^j - 1)^j, {j, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 08 2020 *)

PARI

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

Formula

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

a(n) ~ c * d^n * n! / sqrt(n), where d = (1 + 2*exp(1/r)) * r^2 = 5.90397654567002180048849478643459748789951071189960380570421209622277299181878..., where r = 0.925556278640887084941460444526398190071550948416... is the root of the equation exp(1/r) * (1 + 1/(r*LambertW(-exp(-1/r)/r))) = -1/2 and c = 0.3270084259351451234078681822585067892123772641092056... - Vaclav Kotesovec, Aug 09 2018, updated Oct 08 2020