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. (Formerly M4967 N2131)

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

Offset

3,1

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.

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

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 *)

Crossrefs

Cf. A001147, A007531, A059366.

Sequence in context: A223337 A110755 A206231 * A284096 A022288 A041432

Adjacent sequences: A001753 A001754 A001755 * A001757 A001758 A001759

Keyword

nonn

Author

N. J. A. Sloane

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

Status

approved