A034430 Convolution of A001147 (double factorial numbers) with itself.

1, 2, 7, 36, 249, 2190, 23535, 299880, 4426065, 74294010, 1397669175, 29123671500, 665718201225, 16560190196550, 445300709428575, 12869793995058000, 397815487883438625, 13095523164781307250, 457362512442763302375, 16890682269050394304500

Offset

0,2

Comments

Old name was "Expand arctan(sqrt(x)*sqrt(x+2))/(sqrt(x)*sqrt(x+2)) and multiply n-th term by 1.3.5...(2n+1)".

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

E.g.f.: 1/(1-x)/sqrt(1-2*x). - Vladeta Jovovic, May 11 2003

a(n) = Integral_{x=-infinity..infinity} x^(2*n+1)*exp(-x^2)*erfi(x/sqrt(2)), with erfi the imaginary error function. - Groux Roland, Mar 26 2011

E.g.f.: d/dx(F(x)^(-1)) where (-1) denotes the compositional inverse and F(x) = sin(x)/(1+sin(x)) = x - 2*x^2/2! + 5*x^3/3! - 16*x^4/4! + .... See A000111. - Peter Bala, Jun 24 2012

E.g.f.: E(x) = 1/sqrt(1-2*x)/(1-x) = (1 + x/(U(0)-x))/(1-x), where U(k) = (2*k+1)*x + (k+1) - (k+1)*(2*k+3)*x/U(k+1); (continued fraction Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Jun 27 2012

G.f.: hypergeom([1,1/2],[],2*x)^2. - Mark van Hoeij, May 16 2013

a(n-1)*n = A233481(n) for n >= 1. - Peter Luschny, Dec 14 2013

D-finite with recurrence: a(n) = (3*n-1)*a(n-1)-(2*n-1)*(n-1)*a(n-2) for n >= 2. - Peter Luschny, Dec 14 2013

a(n) ~ 2^(n+3/2) * n^n / exp(n). - Vaclav Kotesovec, Dec 20 2013

a(n) = 2*Pochhammer(1/2, n+1)*hyper2F1([1/2, -n], [3/2], -1). - Peter Luschny, Aug 02 2014

a(n) = -(2*n+1)!! * 2^(-n-1) * Im(Beta(2, n+1, 1/2)). - Vladimir Reshetnikov, Apr 23 2016

Expansion of square of continued fraction 1/(1 - x/(1 - 2*x/(1 - 3*x/(1 - 4*x/(1 - 5*x/(1 - ...)))))). - Ilya Gutkovskiy, Apr 19 2017

Maple

A034430 := proc(n) option remember; if n=0 then 1 elif n=1 then 2 else
(3*n-1)*A034430(n-1)-(1+2*n^2-3*n)*A034430(n-2) fi end: seq(A034430(n), n=0..19); # Peter Luschny, Dec 14 2013

Mathematica

Range[0, 19]! * CoefficientList[Series[1/(1 - x)/Sqrt[1 - 2*x], {x, 0, 19}], x] (* David Scambler, May 24 2012 *)

Crossrefs

Cf. A000111, A001147, A233481.

Sequence in context: A167199 A007889 A125033 * A143805 A249637 A259793

Adjacent sequences: A034427 A034428 A034429 * A034431 A034432 A034433

Keyword

nonn

Author

Jim FitzSimons (cherry(AT)neta.com)

Extensions

Better name from Philippe Deléham, Mar 21 2005

Status

approved