OFFSET
0,3
LINKS
Robert Israel, Table of n, a(n) for n = 0..366
Travis Sherman, Summation of Glaisher- and Apery-like Series, University of Arizona, May 23 2000, p. 11, (3.58) - (3.62).
FORMULA
a(n) = (f1(n)/(2*n-1))*(2*Product_{k=0..n-1} (2*k+1)), where f1(n) corresponds to the x values such that Sum_{k>=0} 2^k/(binomial(2*k,k)*(2*k+(2*n-1))) = x*Pi + y. (See examples for connection with a(n) in terms of material at Links section).
From Robert Israel, Apr 12 2018: (Start)
3*a(n+2) - (18*n+9)*a(n+1) + 12*n*(2*n-1)*a(n) = 0.
E.g.f.: (1-sqrt(1-2*x))/2 + (arctanh(sqrt(2-4*x)) - arctanh(sqrt(2)))*sqrt(2)/4. (End)
a(n) = (n-1)!*2^(n-1)*Sum_{i=0..n-1} binomial(n-3/2, i) for n > 0 with a(0) = 0. - Mikhail Kurkov, Nov 22 2024
a(n) ~ sqrt(Pi) * n^(n - 1/2) * 2^(2*n-2) / exp(n). - Vaclav Kotesovec, Nov 22 2024
EXAMPLE
Examples ((3.58) - (3.62)) at page 11 in Links section as follows, respectively.
For n=1, f1(1) = 1/2, so a(1) = 1.
For n=2, f1(2) = 3/2, so a(2) = 3.
For n=3, f1(3) = 23/6, so a(3) = 23.
For n=4, f1(4) = 91/10, so a(4) = 273.
For n=5, f1(5) = 1451/70, so a(5) = 4353.
MAPLE
P:= 1: A[0]:= 0:
for n from 1 to 20 do
P:= P*(2*(n-1)-3);
A[n]:= 4*(n-1)*A[n-1]-P/3;
od:
seq(A[i], i=0..20); # Robert Israel, Apr 12 2018
MATHEMATICA
RecurrenceTable[{a[n] == 4*(n-1)*a[n-1] - Product[2*k-3, {k, 0, n-1}]/3, a[0] == 0}, a, {n, 0, 20}] (* Vaclav Kotesovec, Apr 02 2018 *)
nmax = 16; Flatten[{0, Table[CoefficientList[Expand[FunctionExpand[Simplify[ Table[Sum[2^j/(Binomial[2*j, j]*(2*j+(2*m-1))), {j, 0, Infinity}]/((2*m-1)/(2*Product[(2*k+1), {k, 0, m-1}])), {m, 1, nmax}]]]], Pi][[n, 2]], {n, 1, nmax}]}] (* Vaclav Kotesovec, Apr 02 2018 *)
PROG
(PARI) a=vector(20); a[1]=1; for(n=2, #a, a[n]=4*(n-1)*a[n-1]-(1/3)*prod(k=0, n-1, (2*k-3))); concat(0, a) \\ Altug Alkan, Apr 01 2018
(Magma) [n le 1 select n else 4*(n-1)*Self(n) - (1/3)*&*[2*k-3: k in [0..n-1]]: n in [0..20]]; // Bruno Berselli, Apr 12 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Detlef Meya, Apr 01 2018
EXTENSIONS
a(10)-a(11) corrected by and more terms from Altug Alkan, Apr 01 2018
STATUS
approved