cp's OEIS Frontend

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

%I A193651 #34 Aug 27 2022 17:47:16
%S A193651 1,2,8,53,473,5198,67568,1013513,17229713,327364538,6874655288,
%T A193651 158117071613,3952926790313,106729023338438,3095141676814688,
%U A193651 95949391981255313,3166329935381425313,110821547738349885938,4100397266318945779688,159915493386438885407813
%N A193651 a(n) = ((2*n + 1)!! + 1)/2.
%C A193651 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 A193651 a(404) has 1002 decimal digits. - _Michael De Vlieger_, Apr 25 2016
%H A193651 Michael De Vlieger, <a href="/A193651/b193651.txt">Table of n, a(n) for n = 0..403</a>
%F A193651 From _Peter Luschny_, Aug 20 2014: (Start)
%F A193651 a(n) = (2^n*Gamma(n+3/2))/sqrt(Pi) + 1/2.
%F A193651 a(n) = 2^n*Pochhammer(1/2, n+1) + 1/2.
%F A193651 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 A193651 (-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 A193651 E.g.f.: (exp(x) + 1/(1-2*x)^(3/2))/2. - _Vladimir Reshetnikov_, Apr 25 2016
%p A193651 seq((1+doublefactorial(2*n+1))/2,n=0..18); # _Peter Luschny_, Aug 20 2014
%t A193651 q[n_, k_] := 1;
%t A193651 r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}]
%t A193651 u[0, x_] := 1; u[n_, x_] := (x + n)*u[n - 1, x]
%t A193651 p[n_, k_] := Coefficient[u[n, x], x, k]
%t A193651 v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
%t A193651 Table[v[n], {n, 0, 18}]    (* A193651 *)
%t A193651 TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
%t A193651 Table[r[k], {k, 0, 8}]  (* 2^k *)
%t A193651 TableForm[Table[p[n, k], {n, 0, 6}, {k, 0, n}]]  (* A130534 *)
%t A193651 Table[((2 n + 1)!! + 1)/2, {n, 0, 18}] (* or *)
%t A193651 Table[(2^n Gamma[n + 3/2])/Sqrt[Pi] + 1/2, {n, 0, 18}] (* or *)
%t A193651 Table[2^n Pochhammer[1/2, n + 1] + 1/2, {n, 0, 18}] (* _Michael De Vlieger_, Apr 25 2016 *)
%o A193651 (Sage)
%o A193651 def A():
%o A193651     n, a, b = 1, 1, 2
%o A193651     yield a
%o A193651     while True:
%o A193651         yield b
%o A193651         n += 1
%o A193651         a, b = b, ((2*(b-a)*n + a)*n - b)/(n-1)
%o A193651 A193651 = A()
%o A193651 [next(A193651) for i in range(19)] # _Peter Luschny_, Aug 20 2014
%Y A193651 Cf. A001147, A193649, A130534.
%K A193651 nonn,easy
%O A193651 0,2
%A A193651 _Clark Kimberling_, Aug 02 2011
%E A193651 New name from _Peter Luschny_, Aug 20 2014