OFFSET
1,2
LINKS
John Tyler Rascoe, Table of n, a(n) for n = 1..100
FORMULA
a(1) = 1; a(n) = 2*(2*n-1)*a(n-1) - (2*n-3)!/(n-1)!.
a(n) = (2*n)!/(4*n!)*(Psi(n+1/2) - Psi(n) + 2*log(2)). - Vladeta Jovovic, Jan 22 2005
E.g.f.: log((sqrt(1-4*x)+1)/2)*(2*x-sqrt(1-4*x)-1)/(-4*x+sqrt(1-4*x)+1). - Vladimir Kruchinin, Mar 17 2016
a(n) = n!*Sum_{k=1..n} binomial(2*n-k-1, n-k)/k. - Vladimir Kruchinin, Mar 17 2016
a(n) ~ log(2) * 2^(2*n - 1/2) * n^n / exp(n). - Vaclav Kotesovec, Mar 17 2016
a(n) = n! * [x^n] -log(1 - x)/(1 - x)^n. - Ilya Gutkovskiy, Sep 21 2017
From Peter Bala, Mar 07 2025: (Start)
a(n+1) = - (2*n+1)!^2/n!^3 * Integral_{x = 0..1} log(x)*x^n*(1 - x)^n = (2*n+1)!^2/n!^3 * Sum_{k = 0..n} (-1)^k * binomial(n, k)/(n+k+1)^2.
(n-1)*a(n) = 2*(4*n^2-10*n+7)*a(n-1) - 4*(n-2)*(2*n-3)^2*a(n-2) with a(1) = 1, a(2) = 5. (End)
EXAMPLE
a(3) = 3!*(1 + (1 + (1 + 1/2)) + (1 + (1 + 1/2) + (1 + 1/2 + 1/3))) = 47.
MAPLE
a := proc(n) option remember; if n = 1 then 1 else 2*(2*n-1)*a(n-1) - (2*n-3)!/(n-1)! fi; end:
seq(a(n), n = 1..20); # Peter Bala, Mar 07 2025
MATHEMATICA
Table[n! Sum[Binomial[2 n - k - 1, n - k]/k, {k, n}], {n, 19}] (* Michael De Vlieger, Mar 17 2016 *)
PROG
(Maxima)
a(n):=n!*sum(binomial(2*n-k-1, n-k)/k, k, 1, n);
/* Vladimir Kruchinin, Mar 17 2016 */
(PARI) lista(nn) = {print1(a=1, ", "); for (n=2, nn, a = a*2*(2*n-1) - (2*n-3)!/(n-1)!; print1(a, ", "); ); } \\ Michel Marcus, Mar 17 2016
CROSSREFS
KEYWORD
easy,nonn,changed
AUTHOR
Leroy Quet, Jan 02 2001
EXTENSIONS
More terms from Michael De Vlieger, Mar 17 2016
STATUS
approved