A224883 -id:A224883 - cp's OEIS frontend

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

Paul D. Hanna, Apr 21 2009

nonn

			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.

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

Cf. A159478, A158093, A183131, A224883.

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

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

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

A246900 Decimal expansion of the constant c = Sum_{n>=0} binomial(n-1 + 1/2^(n-1), n).

Original entry on oeis.org

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

Offset: 1

Paul D. Hanna, Nov 29 2014

			c = 2.55500248436101360804704969796239525102504151483916927730...
where the constant is equal to the sum
c = 1 + binomial(1,1) + binomial(3/2,2) + binomial(9/4,3) + binomial(25/8,4) + binomial(65/16,5) + binomial(161/32,6) +...+ binomial(n-1 + 1/2^(n-1), n) +...
which may be written as
c = 1 + 2/2 + 6/2^4 + 60/2^9 + 2550/2^16 + 476476/2^25 + 384115732/2^36 + 1305385229720/2^49 + 18382187112952806/2^64 +...+ A224883(n)*x^n/2^(n^2) +...
The constant also equals the logarithmic sum
c = 1 + 2*log(2) + 4*log(4/3)^2/2! + 8*log(8/7)^3/3! + 16*log(16/15)^4/4! + 32*log(32/31)^5/5! + 64*log(64/63)^6/6! +...+ (-2)^n*log(1 - 1/2^n)^n/n! +...
which converges rather quickly.

Paul D. Hanna, Table of n, a(n) for n = 1..1001

Cf. A224883.

/* By definition: */
\p128
{c=suminf(n=0,binomial(n-1 + 1/2^(n-1), n)*1.)}
{a(n)=floor(10^n*c)%10}
for(n=0,120,print1(a(n),", "))

/* By a logarithmic identity (accelerated series): */
\p1024
{c=1+suminf(n=1, (-2)^n*log(1 - 1/2^n)^n / n!)}
{a(n)=floor(10^n*c)%10}
for(n=0,1000,print1(a(n),", "))

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

c = Sum_{n>=0} A224883(n) / 2^(n^2), where A224883(n) = (2^n/n!) * Product_{k=0..n-1} (2^(n-1)*k + 1).

cp's OEIS Frontend

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).

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A246900 Decimal expansion of the constant c = Sum_{n>=0} binomial(n-1 + 1/2^(n-1), n).

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