The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A224883 a(n) = 2^(n^2) * binomial(n-1 + 1/2^(n-1), n). 3
1, 2, 6, 60, 2550, 476476, 384115732, 1305385229720, 18382187112952806, 1060603038396055882860, 248959068848694059131153020, 236689359381076468102847994171880, 908758498534088142521911865612937786108, 14063550492706544341683006937639901739122886616 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
FORMULA
G.f.: Sum_{n>=0} (-2)^n * log(1 - x/2^n)^n/n! = Sum_{n>=0} a(n)*x^n/2^(n^2).
a(n) = (2^n/n!) * Product_{k=0..n-1} (2^(n-1)*k + 1).
a(n) = [x^n] 1/(1 - 2^n*x)^(2/2^n).
EXAMPLE
G.f.: A(x) = 1 + 2*x/2 + 6*x^2/2^4 + 60*x^3/2^9 + 2550*x^4/2^16 + 476476*x^5/2^25 +...+ a(n)*x^n/2^(n^2) +...
where
A(x) = 1 - 2*log(1-x/2) + 4*log(1-x/4)^2/2! - 8*log(1-x/8)^3/3! + 16*log(1-x/16)^4/4! +...+ (-2)^n*log(1-x/2^n)^n/n! +...
Illustrate a(n) = [x^n] 1/(1 - 2^n*x)^(2/2^n):
(1-x)^(-2/1) = (1) + 2*x + 3*x^2 + 4*x^3 + 5*x^4 + 6*x^5 +...
(1-2*x)^(-2/2) = 1 + (2)*x + 4*x^2 + 8*x^3 + 16*x^4 + 32*x^5 +...
(1-4*x)^(-2/4) = 1 + 2*x + (6)*x^2 + 20*x^3 + 70*x^4 + 252*x^5 +...
(1-8*x)^(-2/8) = 1 + 2*x + 10*x^2 + (60)*x^3 + 390*x^4 + 2652*x^5 +...
(1-16*x)^(-2/16) = 1 + 2*x + 18*x^2 + 204*x^3 + (2550)*x^4 + 33660*x^5 +...
(1-32*x)^(-2/32) = 1 + 2*x + 34*x^2 + 748*x^3 + 18326*x^4 + (476476)*x^5 +...
where the coefficients in parenthesis form the initial terms of this sequence.
Particular values.
A(1) = 1 + 2*log(2) + 4*log(4/3)^2/2! + 8*log(8/7)^3/3! + 16*log(16/15)^4/4! +...
A(1/2) = 1 + 2*log(4/3) + 4*log(8/7)^2/2! + 8*log(16/15)^3/3! +...
A(1/4) = 1 + 2*log(8/7) + 4*log(16/15)^2/2! + 8*log(32/31)^3/3! +...
A(3/2) = 1 + 2*log(4) + 4*log(8/5)^2/2! + 8*log(16/13)^3/3! + 16*log(32/29)^4/4! +...
Explicitly,
A(1) = 2.55500248436101360804704969796239525102504151...
A(1/2) = 1.61138451105646219391156983544059555709337920...
A(1/4) = 1.27543593708175757392940597050033002345086132...
A(3/2) = 4.22639446385430649517540615961613624264078875...
MATHEMATICA
Table[2^(n^2) Binomial[n-1+1/2^(n-1), n], {n, 0, 20}] (* Harvey P. Dale, Feb 01 2017 *)
PROG
(PARI) {a(n)=2^(n^2)*binomial(n-1+1/2^(n-1), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=(2^n/n!)*prod(k=0, n-1, 2^(n-1)*k + 1)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=2^(n^2)*polcoef(sum(k=0, n, (-2)^k*log(1-x/2^k +x*O(x^n))^k/k!), n)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
Sequence in context: A156472 A108640 A084971 * A001577 A156503 A077175
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 23 2013
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 15 22:50 EDT 2024. Contains 373412 sequences. (Running on oeis4.)