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A034430 Convolution of A001147 (double factorial numbers) with itself. 12

%I

%S 1,2,7,36,249,2190,23535,299880,4426065,74294010,1397669175,

%T 29123671500,665718201225,16560190196550,445300709428575,

%U 12869793995058000,397815487883438625,13095523164781307250,457362512442763302375,16890682269050394304500

%N Convolution of A001147 (double factorial numbers) with itself.

%C 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)".

%H Vincenzo Librandi, <a href="/A034430/b034430.txt">Table of n, a(n) for n = 0..200</a>

%F E.g.f.: 1/(1-x)/sqrt(1-2*x). - _Vladeta Jovovic_, May 11 2003

%F 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

%F 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

%F 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

%F G.f.: hypergeom([1,1/2],[],2*x)^2. - _Mark van Hoeij_, May 16 2013

%F a(n-1)*n = A233481(n) for n >= 1. - _Peter Luschny_, Dec 14 2013

%F 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

%F a(n) ~ 2^(n+3/2) * n^n / exp(n). - _Vaclav Kotesovec_, Dec 20 2013

%F a(n) = 2*Pochhammer(1/2, n+1)*hyper2F1([1/2, -n], [3/2], -1). - _Peter Luschny_, Aug 02 2014

%F a(n) = -(2*n+1)!! * 2^(-n-1) * Im(Beta(2, n+1, 1/2)). - _Vladimir Reshetnikov_, Apr 23 2016

%F 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

%p A034430 := proc(n) option remember; if n=0 then 1 elif n=1 then 2 else

%p (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

%t Range[0, 19]! * CoefficientList[Series[1/(1 - x)/Sqrt[1 - 2*x], {x, 0, 19}], x] (* _David Scambler_, May 24 2012 *)

%Y Cf. A000111, A001147, A233481.

%K nonn

%O 0,2

%A Jim FitzSimons (cherry(AT)neta.com)

%E Better name from _Philippe Deléham_, Mar 21 2005

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Last modified February 23 07:11 EST 2020. Contains 332159 sequences. (Running on oeis4.)