A182525 a(n) = n! * Sum_{k=0..n} binomial(2*n, 2*k) / binomial(n,k).

1, 2, 10, 72, 664, 7440, 98064, 1486464, 25476480, 487192320, 10284768000, 237574149120, 5960907832320, 161440734873600, 4694193123379200, 145855192928256000, 4822943651308339200, 169104439543534387200, 6266811206473703424000

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Sela Fried and Toufik Mansour, Graph labelings obtainable by random walks, arXiv:2304.05728 [math.CO], 2023.

Formula

Asymptotic: a(n) ~ Pi*n^(n+1)*2^(n-1/2)/exp(n). [Vaclav Kotesovec, May 06 2012]

E.g.f.: ( x * arctan(x / sqrt(1 - 2*x)) + sqrt(1 - 2*x) ) / (sqrt(1 - 2*x))^3. -Sela Fried, Apr 26 2023

Example

E.g.f.: A(x) = 1 + 2*x + 10*x^2/2! + 72*x^3/3! + 664*x^4/4! + 7440*x^5/5! +...
a(1) = 1!*(1/1 + 1/1) = 2;
a(2) = 2!*(1/1 + 6/2 + 1/1) = 10;
a(3) = 3!*(1/1 + 15/3 + 15/3 + 1/1) = 72;
a(4) = 4!*(1/1 + 28/4 + 70/6 + 28/4 + 1/1) = 664;
a(5) = 5!*(1/1 + 45/5 + 210/10 + 210/10 + 45/5 + 1/1) = 7440; ...

Mathematica

RecurrenceTable[{a[n]==3*n*a[n-1]-n*(2n-3)*a[n-2], a[0]==1, a[1]==2}, a, {n, 0, 25}] (* Vaclav Kotesovec, May 06 2012 *)

Prog

(PARI) {a(n)=n!*sum(k=0, n, binomial(2*n, 2*k)/binomial(n, k))}

Crossrefs

Sequence in context: A111554 A177384 A354288 * A321389 A292406 A185183

Adjacent sequences: A182522 A182523 A182524 * A182526 A182527 A182528

Keyword

nonn

Author

Paul D. Hanna, May 03 2012

Status

approved