A002690 a(n) = (n+1) * (2*n)! / n!.

1, 4, 36, 480, 8400, 181440, 4656960, 138378240, 4670265600, 176432256000, 7374868300800, 337903056691200, 16838835658444800, 906706535454720000, 52459449551308800000, 3245491278907637760000, 213796737998040637440000, 14940619102451310428160000, 1103945744792235714969600000

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Coefficients of orthogonal polynomials.

E.g.f. for series with alternating signs: x/(1+4*x)^(1/2).

Central terms of triangle A245334. - Reinhard Zumkeller, Aug 30 2014

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• H. E. Salzer, Orthogonal polynomials arising in the evaluation of inverse Laplace transforms, Math. Comp. 9 (1955), 164-177.
• H. E. Salzer, Orthogonal polynomials arising in the evaluation of inverse Laplace transforms, Math. Comp. 9 (1955), 164-177. [Annotated scanned copy]

Crossrefs

a(n) = (n+1) * A001813(n) = 2^n * A001193(n+1).

Cf. A002691, A000407.

Cf. A245334.

Programs

Haskell

a002690 n = a245334 (2 * n) n  -- Reinhard Zumkeller, Aug 30 2014

Magma

[(n+1) * Factorial(2*n) /Factorial(n): n in [0..20]]; // Vincenzo Librandi, Sep 05 2011

Maple

with(combstruct):bin := {B=Union(Z,Prod(B,B))}:
seq (count([B,bin,labeled],size=n+1)*(n+1), n=0..17); # Zerinvary Lajos, Dec 05 2007
A002690 := n -> 2^n*n!*JacobiP(n, -1/2, -n+1, 3):
seq(simplify(A002690(n)), n = 0..18);  # Peter Luschny, Jan 22 2025

Mathematica

Table[((n+1)(2n)!)/n!,{n,0,20}] (* Harvey P. Dale, Sep 04 2011 *)

PARI

a(n)=(n+1)*(2*n)!/n!

Formula

E.g.f.: (1-2*x)/(1-4*x)^(3/2).

a(n) = 2^n*n!*JacobiP(n, -1/2, -n+1, 3). - Peter Luschny, Jan 22 2025

Extensions

Edited by Ralf Stephan, Mar 21 2004