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A159478 a(n) = 2^(n^2+n) * C(1/2^n, n). 5
1, 2, -6, 140, -14570, 6283452, -11049839724, 78893138035608, -2282580118745565210, 267227101453296251927660, -126415241162450125116966673796, 241332381844862786094865482962203112, -1857025703922208959523779453799872508349700 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Sum_{n>=0} C(1/2^n, n) = 1.4306345243611686570661803375590... (A139823).
LINKS
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
KEYWORD
sign
AUTHOR
Paul D. Hanna, Apr 19 2009
STATUS
approved

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Last modified June 20 10:31 EDT 2024. Contains 373516 sequences. (Running on oeis4.)