cp's OEIS Frontend

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.

%I A001756 M4967 N2131 #45 May 13 2020 08:50:09
%S A001756 15,60,450,4500,55125,793800,13097700,243243000,5016886875,
%T A001756 113716102500,2808787731750,75071235739500,2158298027510625,
%U A001756 66409170077250000,2177272076104125000,75769068248423550000,2789248824895091934375,108288483790044745687500
%N 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.
%C A001756 Previous name was: Expansion of an integral.
%D A001756 L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 166-167.
%D A001756 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001756 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001756 T. D. Noe, <a href="/A001756/b001756.txt">Table of n, a(n) for n = 3..100</a>
%H A001756 Louis Comtet, <a href="https://www.jstor.org/stable/43667287">Fonctions génératrices et calcul de certaines intégrales</a>, Publikacije Elektrotechnickog faculteta - Serija Matematika i Fizika, No. 181/196 (1967), 77-87; see p. 85.
%F A001756 a(n) = 5*A007531(n)*A001147(n-2)/(2*(2*n-5)). - _Philippe Deléham_, Jun 26 2006
%F A001756 a(3) = 15, a(n) = a(n-1)*n*(2*n-7)/(n-3). - _Philippe Deléham_, Sep 19 2006
%F A001756 From _Petros Hadjicostas_, May 12 2020: (Start)
%F A001756 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]
%F A001756 a(n) = 20 * binomial(2*n-6, n-3) * n!/2^n for n >= 3. [Special case of a formula due to _Reinhard Zumkeller_]
%F A001756 a(n) = binomial(-1/2, 3) * binomial(-1/2, n-3) * (-1)^n * n! * 2^n for n >= 3. (End)
%F A001756 a(n) ~ sqrt(2)*(5/16)*(2*n/e)^n. - _Peter Luschny_, May 13 2020
%t A001756 RecurrenceTable[{a[3]==15,a[n]==a[n-1]n (2n-7)/(n-3)},a,{n,20}] (* _Harvey P. Dale_, Nov 08 2011 *)
%t A001756 Join[{c = 15}, Table[c = c*n*(2*n - 7)/(n - 3), {n, 4, 20}]] (* _T. D. Noe_, Aug 10 2012 *)
%Y A001756 Cf. A001147, A007531, A059366.
%K A001756 nonn
%O A001756 3,1
%A A001756 _N. J. A. Sloane_
%E A001756 More terms from _Philippe Deléham_, Sep 19 2006
%E A001756 Corrected and extended by _Harvey P. Dale_, Nov 08 2011
%E A001756 New name by _Petros Hadjicostas_, May 12 2020