A159478 a(n) = 2^(n^2+n) * C(1/2^n, n).

1, 2, -6, 140, -14570, 6283452, -11049839724, 78893138035608, -2282580118745565210, 267227101453296251927660, -126415241162450125116966673796, 241332381844862786094865482962203112, -1857025703922208959523779453799872508349700

Offset

0,2

Comments

Sum_{n>=0} C(1/2^n, n) = 1.4306345243611686570661803375590... (A139823).

Links

G. C. Greubel, Table of n, a(n) for n = 0..57

Formula

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

a(n) = [x^n] (1 + 2^(n+1)*x)^(1/2^n).

a(n) ~ -(-1)^n * 2^(n^2)/n. - Vaclav Kotesovec, Jun 29 2018

Example

G.f.: A(x) = 1 +2*x/2^2 -6*x^2/2^6 +140*x^3/2^12 -14570*x^4/2^20 +...
A(x) = 1 + log(1+x/2) + log(1+x/4)^2/2! + log(1+x/8)^3/3! +...
Illustrate a(n) = [x^n] (1 + 2^(n+1)*x)^(1/2^n):
(1+4*x)^(1/2) = 1 + (2)*x - 2*x^2 + 4*x^3 - 10*x^4 +...
(1+8*x)^(1/4) = 1 + 2*x - (6)*x^2 + 28*x^3 - 154*x^4 +...
(1+16*x)^(1/8) = 1 + 2*x - 14*x^2 + (140)*x^3 - 1610*x^4 +...
(1+32*x)^(1/16) = 1 + 2*x - 30*x^2 + 620*x^3 - (14570)*x^4 +...
(1+64*x)^(1/32) = 1 + 2*x - 62*x^2 + 2604*x^3 - 123690*x^4 + (6283452)*x^5 +...

Mathematica

Table[2^(n^2 + n)*Binomial[1/2^n, n], {n, 0, 25}] (* G. C. Greubel, Jun 12 2018 *)

Prog

(PARI) a(n)=2^(n^2+n)*binomial(1/2^n, n)
(Magma) SetDefaultRealField(RealField(250)); [Round(2^(n + n^2)*Gamma(1 + 1/2^n)/(Gamma(n+1)*Gamma(1 + 1/2^n - n))): n in [0..25]]; // G. C. Greubel, Jun 12 2018

Crossrefs

Cf. A139823.

Sequence in context: A206849 A368395 A090907 * A047937 A027731 A280821

Adjacent sequences: A159475 A159476 A159477 * A159479 A159480 A159481

Keyword

sign

Author

Paul D. Hanna, Apr 19 2009

Status

approved