A002690 a(n) = (n+1) * (2*n)! / n!. (Formerly M3665 N1491)

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

Offset

0,2

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]

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

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 *)

Prog

(PARI) a(n)=(n+1)*(2*n)!/n!
(Magma) [(n+1) * Factorial(2*n) /Factorial(n): n in [0..20]]; // Vincenzo Librandi, Sep 05 2011
(Haskell)
a002690 n = a245334 (2 * n) n  -- Reinhard Zumkeller, Aug 30 2014

Crossrefs

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

Cf. A002691, A000407.

Cf. A245334.

Sequence in context: A135906 A197446 A291313 * A370927 A094417 A349504

Adjacent sequences: A002687 A002688 A002689 * A002691 A002692 A002693

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

Edited by Ralf Stephan, Mar 21 2004

Status

approved