OFFSET
1,2
COMMENTS
Equals central terms of A174449. - G. C. Greubel, Nov 29 2021
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 54 [Link is broken.]
FORMULA
E.g.f.: ((1/2)/x)*(1-sqrt(1-4*x^2)). [With interspersed zeros.]
Recurrence: b(1)=1, b(2)=0, b(n)=(4*n^3-12*n^2+8*n)*b(n-2)/(n+1) and a(n) = b(2*n-1).
a(n) = (2n-1)/n * ( (2(n-1))! / (n-1)! )^2. - Travis Kowalski (kowalski(AT)euclid.UCSD.Edu), Dec 15 2000
i*sin(arcsec(2*x)) = -1/(2*x) + x + 6*x^3/3! + 240*x^5/5! + 25200*x^7/7! + ...
a(n) = 2^(n-1) * A036770(n).
a(n) = (2*n-1)! * A000108(n-1). - Michail Stamatakis, Jan 24 2019
Sum_{n>=1} 1/a(n) = 1 + StruveL(0, 1/2)*Pi/8 + StruveL(1, 1/2)*Pi/4, where StruveL is the modified Struve function. - Amiram Eldar, Dec 04 2022
MAPLE
spec := [S, {S=Union(Z, Prod(Z, S, S))}, labeled]:
seq(combstruct[count](spec, size=2*n-1), n=1..14);
# second Maple program:
a:= proc(n) option remember; `if`(n<2, n,
4*(n-1)*(2*n-3)*(2*n-1)*a(n-1)/n)
end:
seq(a(n), n=1..22); # Alois P. Heinz, Dec 03 2019
MATHEMATICA
nn=20; f[x_]:=Sum[a[n]x^n/n!, {n, 0, nn}]; s=SolveAlways[0==Series[f[x]-x(1+f[x]^2), {x, 0, nn}], x]; Select[Flatten[Table[a[n], {n, 0, nn}]/.s], #>0&] (* Geoffrey Critzer, Mar 23 2013 *)
RecurrenceTable[{a[1]==1, n*a[n]==(4*(n-1)*(2*n-3)*(2*n-1))*a[n-1]}, a[n], {n, 1, 22}] (* Georg Fischer, Dec 03 2019 following Alois P. Heinz *)
a[n_]:= CatalanNumber[n-1] Gamma[2n]; Array[a, 14] (* Peter Luschny, Dec 03 2019 *)
PROG
(PARI) a=vector(28); print1(a[1]=1, ", "); forstep(k=1, #a-2, 2, print1(a[k+2]=4*a[k]*(k^3+3*k^2+2*k)/(k+3), ", ")) \\ Hugo Pfoertner, Dec 04 2019
(Sage) [factorial(2*n-1)*catalan_number(n-1) for n in (1..15)] # G. C. Greubel, Nov 29 2021
(Magma) [Factorial(2*n-1)*Catalan(n-1): n in [1..15]]; // G. C. Greubel, Nov 29 2021
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
EXTENSIONS
Edited by Georg Fischer, Dec 03 2019
STATUS
approved