The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A159318 a(n) = 2^(n^2+n) * binomial(2*n-1 + 1/2^n, n) / (n*2^n + 1). 3
 1, 2, 26, 1804, 591894, 860081340, 5338683113364, 138637536961147800, 14872932935424544987110, 6538678365573711555851779180, 11717380780236748297970244719026812 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS G. C. Greubel, Table of n, a(n) for n = 0..56 FORMULA a(n) = [x^n] {(1-sqrt(1 - 2^(n+3)*x))/(2^(n+2)*x)}^(1/2^n). G.f.: A(x) = Sum_{n>=0} a(n)*x^n/2^(n^2+n). G.f.: A(x) = Sum_{n>=0} log(F(x/2^n))^n/n! where F(x) = (1-sqrt(1-4*x))/(2*x) is the Catalan function (A000108). Radius of convergence of A(x) is |x| <= 1/2. a(n) = [x^n] (1/(1 - 2^(n+1)*x)^(n + 1/2^n))/(n*2^n + 1). - Paul D. Hanna, Jun 15 2010 EXAMPLE G.f.: A(x) = 1 + 2*x/2^2 + 26*x^2/2^6 + 1804*x^3/2^12 + 591894*x^4/2^20 + ... G.f.: A(x) = Sum_{n>=0} log( 2^n*(1-sqrt(1 - 4*x/2^n))/(2*x) )^n/n!. A(x) = 1 + log(F(x/2)) + log(F(x/4))^2/2! + log(F(x/8))^3/3! + ... where F(x) = (1-sqrt(1 - 4*x))/(2*x). Special values. A(1/2) = 1 + log(2) + log(4-4*sqrt(1/2))^2/2! + log(8-8*sqrt(3/4))^3/3! + log(16-16*sqrt(7/8))^4/4! + ... A(1/2) = 1.70573970062357248928512380703308976974285275... A(-1/2) = 1 + log(2*sqrt(2)-2) + log(4*sqrt(3/2)-4)^2/2! + log(8*sqrt(5/4)-8)^3/3! + log(16*sqrt(9/8)-16)^4/4! + ... A(-1/2) = 0.81741280310249092844743171863299249334671633... Illustrate a(n) = [x^n] {(1-sqrt(1-2^(n+3)*x))/(2^(n+2)*x)}^(1/2^n): n=0: (1) + 2*x + 8*x^2 + 40*x^3 + 224*x^4 + 1344*x^5 + ... n=1: 1 + (2)*x + 14*x^2 + 132*x^3 + 1430*x^4 + 16796*x^5 + ... n=2: 1 + 2*x + (26)*x^2 + 476*x^3 + 10150*x^4 + 236060*x^5 + ... n=3: 1 + 2*x + 50*x^2 + (1804)*x^3 + 76342*x^4 + 3534076*x^5 + ... n=4: 1 + 2*x + 98*x^2 + 7020*x^3 + (591894)*x^4 + 54673468*x^5 + ... n=5: 1 + 2*x + 194*x^2 + 27692*x^3 + 4660950*x^4 + (860081340)*x^5 + ... coefficients in parenthesis form the initial terms of this sequence. MATHEMATICA Table[2^(n^2 +n)*Binomial[2*n -1 +1/2^n, n]/(n*2^n +1), {n, 0, 50}] (* G. C. Greubel, Jun 26 2018 *) PROG (PARI) a(n)=2^(n^2+n)*binomial(2*n-1+1/2^n, n)/(n*2^n + 1) (PARI) a(n)=polcoeff(((1-sqrt(1 - 2^(n+3)*x))/2^(n+2)/x)^(1/2^n), n) (PARI) {a(n)=polcoeff(1/(1-2^(n+1)*x+x*O(x^n))^(n+1/2^n), n)/(n*2^n+1)} \\ Paul D. Hanna, Jun 15 2010 (MAGMA) [2^(n^2 +n)*Binomial(2*n -1 +1/2^n, n)/(n*2^n +1): n in [0..50]]; // G. C. Greubel, Jun 26 2018 CROSSREFS Cf. A159558, A159478, A158093, A000108. Sequence in context: A209916 A337578 A156213 * A318132 A134795 A268667 Adjacent sequences:  A159315 A159316 A159317 * A159319 A159320 A159321 KEYWORD nonn AUTHOR Paul D. Hanna, Apr 22 2009 STATUS approved

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

Last modified December 2 08:31 EST 2021. Contains 349437 sequences. (Running on oeis4.)