A001756 a(n) = A059366(n,n-3) = A059366(n,3) for n >= 3, where the triangle A059366 arises from the expansion of a trigonometric integral.

15, 60, 450, 4500, 55125, 793800, 13097700, 243243000, 5016886875, 113716102500, 2808787731750, 75071235739500, 2158298027510625, 66409170077250000, 2177272076104125000, 75769068248423550000, 2789248824895091934375, 108288483790044745687500

Offset: 3

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Previous 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

• T. D. Noe, Table of n, a(n) for n = 3..100
• 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.

Crossrefs

Cf. A001147, A007531, A059366.

Programs

Mathematica

RecurrenceTable[{a[3]==15,a[n]==a[n-1]n (2n-7)/(n-3)},a,{n,20}] (* Harvey P. Dale, Nov 08 2011 *)
Join[{c = 15}, Table[c = c*n*(2*n - 7)/(n - 3), {n, 4, 20}]] (* T. D. Noe, Aug 10 2012 *)

Formula

a(n) = 5*A007531(n)*A001147(n-2)/(2*(2*n-5)). - Philippe Deléham, Jun 26 2006

a(3) = 15, a(n) = a(n-1)*n*(2*n-7)/(n-3). - Philippe Deléham, Sep 19 2006

From Petros Hadjicostas, May 12 2020: (Start)

a(n) = n! * Sum_{k=0..n-3} (-1)^k * 2^(2*k-n) * binomial(n-3, k) * binomial(2*n-2*k, n-3) * binomial(n-2*k+3, n-k) for n >= 3. [Special case of a formula by Comtet, but corrected]

a(n) = 20 * binomial(2*n-6, n-3) * n!/2^n for n >= 3. [Special case of a formula due to Reinhard Zumkeller]

a(n) = binomial(-1/2, 3) * binomial(-1/2, n-3) * (-1)^n * n! * 2^n for n >= 3. (End)

a(n) ~ sqrt(2)*(5/16)*(2*n/e)^n. - Peter Luschny, May 13 2020

Extensions

More terms from Philippe Deléham, Sep 19 2006

Corrected and extended by Harvey P. Dale, Nov 08 2011

New name by Petros Hadjicostas, May 12 2020