A175925 - OEIS

%I #24 Sep 08 2022 08:45:52

%S 1,6,30,168,1080,7920,65520,604800,6168960,68947200,838252800,

%T 11017036800,155675520000,2353813862400,37922556672000,

%U 648606486528000,11737685127168000,224083079700480000,4500868715126784000

%N a(n) = (2*n+1)*(n+1)!.

%C The denominators of the Taylor expansion coefficients of the double integral d(u) = int_0^1 dx int_0^1 dy exp(-u^2*(x-y)^2) = Sum_{n>=0} (-1)^n*u^(2n)/a(n).

%H Vincenzo Librandi, <a href="/A175925/b175925.txt">Table of n, a(n) for n = 0..400</a>

%H D. H. Bailey, J. M. Borwein, R. E. Crandall, <a href="http://dx.doi.org/10.1090/S0025-5718-10-02338-0">Advances in the theory of box integrals</a>, Math. Comp. 79 (271) (2010) 1839-1866, eq (18).

%H Milan Janjić, <a href="https://www.emis.de/journals/JIS/VOL21/Janjic2/janjic103.html">Pascal Matrices and Restricted Words</a>, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.

%F a(n) = A005408(n)*A000142(n+1) = (n+1)*A007680(n).

%F E.g.f.: (1 + 3*x)/(1 - x)^3. - _Ilya Gutkovskiy_, May 12 2017

%F From _Amiram Eldar_, Aug 04 2020: (Start)

%F Sum_{n>=0} 1/a(n) = sqrt(Pi)*erfi(1) + 1 - e.

%F Sum_{n>=0} (-1)^n/a(n) = sqrt(Pi)*erf(1) - 1 + 1/e. (End)

%p A := proc(n) (2*n+1)*(n+1)! ; end proc:

%t Table[(2n+1)(n+1)!,{n,0,20}] (* _Harvey P. Dale_, Sep 30 2011 *)

%o (Magma) [(2*n+1)*Factorial(n+1): n in [0..20]]; // _Vincenzo Librandi_, Oct 11 2011

%Y Cf. A000142, A005408, A007680.

%K nonn,easy

%O 0,2

%A _R. J. Mathar_, Oct 19 2010