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A193651 a(n) = ((2*n + 1)!! + 1)/2. 4

%I

%S 1,2,8,53,473,5198,67568,1013513,17229713,327364538,6874655288,

%T 158117071613,3952926790313,106729023338438,3095141676814688,

%U 95949391981255313,3166329935381425313,110821547738349885938,4100397266318945779688

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

%C Previous name was: Q-residue of the triangle A130534, where Q is the triangular array (t(i,j)) given by t(i,j)=1. For the definition of Q-residue, see A193649.

%C a(404) has 1002 decimal digits. - _Michael De Vlieger_, Apr 25 2016

%H Michael De Vlieger, <a href="/A193651/b193651.txt">Table of n, a(n) for n = 0..403</a>

%F From _Peter Luschny_, Aug 20 2014 : (Start)

%F a(n) = (2^n*Gamma(n+3/2))/sqrt(Pi) + 1/2.

%F a(n) = 2^n*pochhammer(1/2, n+1) + 1/2.

%F a(n) = ((2*a(n-1) - 2*a(n-2))*n^2 + a(n-2)*n - a(n-1))/(n-1) for n>1, a(0)=1, a(1)=2. (End)

%F (-n+1)*a(n) +(2*n^2-1)*a(n-1) -n*(2*n-1)*a(n-2)=0. - _R. J. Mathar_, Feb 19 2015

%F E.g.f.: (exp(x) + 1/(1-2*x)^(3/2))/2. - _Vladimir Reshetnikov_, Apr 25 2016

%p seq((1+doublefactorial(2*n+1))/2,n=0..18); # _Peter Luschny_, Aug 20 2014

%t q[n_, k_] := 1;

%t r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}]

%t u[0, x_] := 1; u[n_, x_] := (x + n)*u[n - 1, x]

%t p[n_, k_] := Coefficient[u[n, x], x, k]

%t v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]

%t Table[v[n], {n, 0, 18}] (* A193651 *)

%t TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]

%t Table[r[k], {k, 0, 8}] (* 2^k *)

%t TableForm[Table[p[n, k], {n, 0, 6}, {k, 0, n}]] (* A130534 *)

%t Table[((2 n + 1)!! + 1)/2, {n, 0, 18}] (* or *)

%t Table[(2^n Gamma[n + 3/2])/Sqrt[Pi] + 1/2, {n, 0, 18}] (* or *)

%t Table[2^n Pochhammer[1/2, n + 1] + 1/2, {n, 0, 18}] (* _Michael De Vlieger_, Apr 25 2016 *)

%o (Sage)

%o def A():

%o n, a, b = 1, 1, 2

%o yield a

%o while True:

%o yield b

%o n += 1

%o a, b = b, ((2*(b-a)*n + a)*n - b)/(n-1)

%o A193651 = A()

%o [next(A193651) for i in range(19)] # _Peter Luschny_, Aug 20 2014

%Y Cf. A001147, A193649, A130534.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Aug 02 2011

%E New name from _Peter Luschny_, Aug 20 2014

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Last modified May 8 13:26 EDT 2021. Contains 343666 sequences. (Running on oeis4.)