A001814 Coefficient of H_2 when expressing x^{2n} in terms of Hermite polynomials H_m. (Formerly M4875 N2088)

1, 12, 180, 3360, 75600, 1995840, 60540480, 2075673600, 79394515200, 3352212864000, 154872234316800, 7771770303897600, 420970891461120000, 24481076457277440000, 1521324036987955200000, 100610229646136770560000

Offset

1,2

Comments

a(n) = A126804(n)/2. - Zerinvary Lajos, Sep 21 2007

References

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 801.

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 = 1..200

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

H. E. Salzer, Coefficients for expressing the first thirty powers in terms of the Hermite polynomials, Math. Comp., 3 (1948), 167-169.

Index entries for sequences related to Hermite polynomials

Formula

E.g.f.: x*(1 + 2x)/(1 - 4x)^(5/2).

a(n) = (2*n)!/(2*(n-1)!).

(n!/2)*binomial(2*n,n)*n or n!/2*A005430. - Zerinvary Lajos, Jun 06 2006

Maple

with(combinat):for n from 1 to 16 do printf(`%d, `, n!/2*sum(binomial(2*n, n), k=1..n)) od: # Zerinvary Lajos, Mar 13 2007
a:=n->sum((count(Permutation(n*2+2), size=n+1)), j=0..n)/2: seq(a(n), n=0..15); # Zerinvary Lajos, May 03 2007
seq(1/2*mul((n+k), k=1..n), n=0..16); # Zerinvary Lajos, Sep 21 2007

Mathematica

Table[(2*n)!/(2*(n-1)!), {n, 1, 20}] (* Vincenzo Librandi, Nov 22 2011 *)

Prog

(MuPAD) combinat::catalan(n)*binomial(n+1, 2)*n! $ n = 1..16; // Zerinvary Lajos, Feb 15 2007
(Magma) [Factorial(2*n)/(2*Factorial(n-1)): n in [1..20]]; // Vincenzo Librandi, Nov 22 2011

Crossrefs

a(n) = A048854(n, 1) = A067147(2n, 2).

Cf. A001879.

Cf. A005430.

Sequence in context: A241710 A318245 A051609 * A370750 A327079 A013924

Adjacent sequences: A001811 A001812 A001813 * A001815 A001816 A001817

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms and new description from Christian G. Bower, Dec 18 2001

Status

approved