A001814 Coefficient of H_2 when expressing x^{2n} in terms of Hermite polynomials H_m.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• changed

Comments

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

a(n) is the number of ways to partition a set of 2n elements into parts of size 2 and then multiply by the number n of parts. - Alain Goupil, Jul 27 2025

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

Crossrefs

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

Cf. A001879.

Cf. A005430.

Programs

Magma

[Factorial(2*n)/(2*Factorial(n-1)): n in [1..20]]; // Vincenzo Librandi, Nov 22 2011

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 *)

MuPAD

combinat::catalan(n)*binomial(n+1,2)*n! $ n = 1..16; // Zerinvary Lajos, Feb 15 2007

Formula

E.g.f.: x/(1 - 4*x)^(3/2). - corrected by Alain Goupil, Jul 28 2025

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

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

Sum_{n>=0} a(n)*x^(2n)/(2n)! = (x^2/2)*exp(x^2). - Alain Goupil, Jul 28 2025

Extensions

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