A139823 -id:A139823 - cp's OEIS frontend

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

Paul D. Hanna, Apr 19 2009

sign

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

			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 +...

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

Cf. A139823.

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

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

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

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

A139824 Decimal expansion of constant c = Sum_{n>=0} C(1/2^n, n)*2^n.

Original entry on oeis.org

1, 7, 7, 7, 2, 0, 8, 0, 1, 6, 1, 1, 9, 6, 9, 1, 6, 2, 6, 2, 3, 1, 1, 2, 5, 1, 9, 7, 2, 4, 4, 0, 3, 5, 3, 3, 1, 2, 3, 8, 0, 0, 6, 4, 1, 2, 5, 1, 1, 2, 6, 3, 1, 2, 3, 3, 2, 2, 9, 0, 6, 6, 6, 3, 2, 1, 6, 2, 0, 0, 5, 5, 8, 7, 8, 9, 7, 5, 2, 3, 4, 5, 6, 7, 6, 6, 4, 2, 8, 5, 4, 1, 7, 8, 9, 6, 5, 9, 4, 7, 5, 0, 4, 5, 1

Offset: 1

Paul D. Hanna, May 01 2008

			c = 1.77720801611969162623112519724403533123800641251126312332290666...
c = 1 + (1/2)*2 - (3/32)*2^2 + (35/1024)*2^3 - (7285/524288)*2^4 +...
c = 1 + log(2) + log(3/2)^2/2! + log(5/4)^3/3! + log(9/8)^4/4! +...
The formulas for this constant illustrate the identity:
Sum_{n>=0} log(1 + q^n*x)^n*y^n/n! = Sum_{n>=0} binomial(q^n*y, n)*x^n.

Cf. A139823, A139825.

a(n)=local(c=sum(m=0,n+2,log(1+2/2^m)^m/m!));floor(c*10^n)%10

c = Sum_{n>=0} log(1 + 2/2^n)^n/n! .

A139825 Decimal expansion of constant c = Sum_{n>=0} C(3/2^n, n).

Original entry on oeis.org

2, 4, 4, 7, 8, 6, 2, 6, 0, 5, 7, 5, 1, 5, 7, 7, 0, 3, 5, 0, 3, 2, 2, 7, 0, 0, 5, 6, 4, 9, 1, 2, 5, 1, 5, 3, 5, 1, 6, 3, 2, 6, 2, 9, 6, 4, 9, 4, 1, 4, 3, 1, 4, 6, 3, 3, 8, 8, 3, 8, 1, 6, 7, 8, 1, 8, 9, 7, 9, 3, 0, 5, 7, 8, 0, 8, 5, 5, 0, 0, 4, 7, 9, 7, 7, 6, 0, 1, 6, 6, 3, 3, 8, 1, 9, 8, 5, 7, 2, 6, 5, 8, 5, 9, 9

Offset: 1

Paul D. Hanna, May 01 2008

			c = 2.44786260575157703503227005649125153516326296494143146338838167...
c = 1 + 3/2 - 3/32 + 65/1024 - 16965/524288 + 4112925/268435456 +...
c = 1 + log(3/2)*3 + log(5/4)^2*3^2/2! + log(9/8)^3*3^3/3! +...
The formulas for this constant illustrate the identity:
Sum_{n>=0} log(1 + q^n*x)^n*y^n/n! = Sum_{n>=0} binomial(q^n*y, n)*x^n.

Cf. A139823, A139824.

a(n)=local(c=sum(m=0,n+2,log(1+1/2^m)^m*3^m/m!));floor(c*10^n)%10

c = Sum_{n>=0} log(1 + 1/2^n)^n*3^n/n! .

cp's OEIS Frontend

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

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A139824 Decimal expansion of constant c = Sum_{n>=0} C(1/2^n, n)*2^n.

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PARI

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A139825 Decimal expansion of constant c = Sum_{n>=0} C(3/2^n, n).

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Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula