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

 Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations (which often say "discovered thanks to the OEIS"). Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A301990 a(n) = 8*(n-1)*a(n-1) + Product_{k=0..n-2} (2*k-1) with a(1) = 1. 1
 1, 7, 111, 2661, 85137, 3405375, 163457055, 9153584685, 585829284705, 42179706471735, 3374376483279375, 296945129873855925, 28506732454140858225, 2964700174914415112175, 332046419582508638982975, 39845570349687578631280125, 5100233004753819781450226625 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS Travis Sherman, Summation of Glaisher- and Apery-like Series, University of Arizona, May 23 2000, p. 10, (3.43) - (3.47). FORMULA a(n) = (f1(n)/(4*n-2))*9*Product_{k=1..n} (2*k-1) where f1(n) corresponds to the x values such that Sum_{k>=0} 1/(binomial(2*k,k)*(2*k+(2*n-1))) = x*Pi*sqrt(3) - y. (See examples for connection with a(n) in terms of material at Links section). EXAMPLE Examples ((3.43) - (3.47)) at page 10 in Links section as follows, respectively. For n=1, f1(1) = 2/9, so a(1) = 1. For n=2, f1(2) = 14/9, so a(2) = 7. For n=3, f1(3) = 74/9, so a(3) = 111. For n=4, f1(4) = 1774/45, so a(4) = 2661. For n=5, f1(5) = 56758/315, so a(5) = 85137. MATHEMATICA RecurrenceTable[{a[n + 1] == 8*n*a[n] + Product[(2*k - 1), {k, 0, n - 1}], a[1] == 1}, a, {n, 1, 20}] (* Vaclav Kotesovec, Mar 30 2018 *) Table[FullSimplify[2^(-4 + 3 n) Sqrt[3] Gamma[n] + 2^(-2 - n) Gamma[-1 + 2 n] Hypergeometric2F1Regularized[1, -1/2 + n, 1 + n, 1/4]], {n, 1, 20}] (* Vaclav Kotesovec, Mar 30 2018 *) nmax = 15; Table[1/Sqrt[3]*CoefficientList[Expand[FunctionExpand[Table[ FullSimplify[Sum[1/(Binomial[2*j, j]*(2*j + (2*m - 1))), {j, 0, Infinity}]] * 9 * Product[(2*k - 1), {k, 1, m}]/(4*m - 2), {m, 1, nmax}]]], Pi][[n, 2]], {n, 2, nmax}] (* Vaclav Kotesovec, Apr 12 2018 *) PROG (PARI) a=vector(20); a[1]=1; for(n=2, #a, a[n]=8*(n-1)*a[n-1] + prod(k=0, n-2, 2*k-1)); a \\ Altug Alkan, Mar 30 2018 CROSSREFS Cf. A000984, A301992. Sequence in context: A212371 A112463 A009471 * A193441 A260027 A061233 Adjacent sequences:  A301987 A301988 A301989 * A301991 A301992 A301993 KEYWORD nonn AUTHOR Detlef Meya, Mar 30 2018 EXTENSIONS More terms from Vaclav Kotesovec, Mar 30 2018 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 5 05:51 EST 2020. Contains 338944 sequences. (Running on oeis4.)