A159558 a(n) = 2^(n^2+n) * C(n-1 + 1/2^n, n) = [x^n] 1/(1 - 2^(n+1)*x)^(1/2^n).

1, 2, 10, 204, 18326, 7157436, 11867138452, 81971848887192, 2329289249771718630, 270079267572894401313900, 127115660247624311548253487740, 242023658005438716992830183038644712

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..57

Formula

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

a(n) ~ 2^(n^2) / n. - Vaclav Kotesovec, Oct 20 2020

Example

G.f.: A(x) = 1 + 2*x/2^2 + 10*x^2/2^6 + 204*x^3/2^12 + 18326*x^4/2^20 +...
A(x) = 1 - log(1-x/2) + log(1-x/4)^2/2! - log(1-x/8)^3/3! +...+ (-1)^n*log(1-x/2^n)^n/n! +...
Illustrate a(n) = [x^n] 1/(1 - 2^(n+1)*x)^(1/2^n):
(1-4*x)^(-1/2) = 1 + (2)*x + 6*x^2 + 20*x^3 + 70*x^4 + 252*x^5 +...
(1-8*x)^(-1/4) = 1 + 2*x + (10)*x^2 + 60*x^3 + 390*x^4 + 2652*x^5 +...
(1-16*x)^(-1/8) = 1 + 2*x + 18*x^2 + (204)*x^3 + 2550*x^4 + 33660*x^5 +...
(1-32*x)^(-1/16) = 1 + 2*x + 34*x^2 + 748*x^3 + (18326)*x^4 + 476476*x^5 +...
(1-64*x)^(-1/32) = 1 + 2*x + 66*x^2 + 2860*x^3 + 138710*x^4 + (7157436)*x^5 +...
where the coefficients in parenthesis form the initial terms of this sequence.
Particular values.
A(1) = 1 + log(2) + log(4/3)^2/2! + log(8/7)^3/3! + log(16/15)^4/4! +...
A(1/2) = 1 + log(4/3) + log(8/7)^2/2! + log(16/15)^3/3! +...
A(1/4) = 1 + log(8/7) + log(16/15)^2/2! + log(32/31)^3/3! +...
A(3/2) = 1 + log(4) + log(8/5)^2/2! + log(16/13)^3/3! + log(32/29)^4/4! +...
Explicitly,
A(1) = 1.734925215983391138169827514899...
A(3/2) = 2.498242012620581570762548014070...
A(r) = 2 at r=1.2139293567161900826815...
A(r) = 3 at r=1.6849757886374480509741...
A(-1) = 0.6191596458119190547682348949108188...
A(-2) = 0.3872099757580366707782339498635620...
A(2) is indeterminate.

Mathematica

Table[2^(n^2+n) * Binomial[n-1+1/2^n, n], {n, 0, 15}] (* Vaclav Kotesovec, Oct 20 2020 *)

Prog

(PARI) a(n)=2^(n^2+n)*binomial(n-1+1/2^n, n)

Crossrefs

Cf. A159478, A158093, A183131, A224883.

Sequence in context: A264563 A156510 A246532 * A297066 A320395 A001528

Adjacent sequences: A159555 A159556 A159557 * A159559 A159560 A159561

Keyword

nonn

Author

Paul D. Hanna, Apr 21 2009

Status

approved