A001194 a(n) = A059366(n,n-2) = A059366(n,2) for n >= 2, where the triangle A059366 arises in the expansion of a trigonometric integral.

3, 9, 54, 450, 4725, 59535, 873180, 14594580, 273648375, 5685805125, 129636356850, 3217338674550, 86331921100425, 2490343877896875, 76844896803675000, 2525635608280785000, 88081541838792376875, 3248654513701342370625

Offset: 2

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Old name was: Expansion of an integral.

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 166-167.
• 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

• Reinis Cirpons, James East, and James D. Mitchell, Transformation representations of diagram monoids, arXiv:2411.14693 [math.RA], 2024. See p. 3.
• Louis Comtet, Fonctions génératrices et calcul de certaines intégrales, Publikacije Elektrotechnickog faculteta - Serija Matematika i Fizika, No. 181/196 (1967), 77-87; see p. 85.

Programs

Mathematica

Table[3*n*(n-1)*(2*n-4)!/(2^(n-1)*(n-2)!),{n,2,20}] (* Vaclav Kotesovec, Jan 05 2014 *)

Formula

a(n) = (2*n - 1)*a(n-1) - 3*(n - 1)*(2*n - 7)!! for n > 3. - Sean A. Irvine, Mar 23 2012

a(n) = 3*n*(n-1)*(2*n-4)!/(2^(n-1)*(n-2)!) for n >= 2. - Vaclav Kotesovec, Jan 05 2014

a(n) = binomial(-1/2, 2) * binomial(-1/2, n-2) * (-1)^n * n! * 2^n for n >= 2. - Petros Hadjicostas, May 13 2020

a(n) ~ sqrt(2)*(3/8)*(2*n/e)^n. - Peter Luschny, May 14 2020

Extensions

More terms from Sean A. Irvine, Mar 22 2012

New name by Petros Hadjicostas, May 13 2020