OFFSET
0,2
COMMENTS
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.
a(404) has 1002 decimal digits. - Michael De Vlieger, Apr 25 2016
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..403
FORMULA
From Peter Luschny, Aug 20 2014: (Start)
a(n) = (2^n*Gamma(n+3/2))/sqrt(Pi) + 1/2.
a(n) = 2^n*Pochhammer(1/2, n+1) + 1/2.
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)
(-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
E.g.f.: (exp(x) + 1/(1-2*x)^(3/2))/2. - Vladimir Reshetnikov, Apr 25 2016
MAPLE
seq((1+doublefactorial(2*n+1))/2, n=0..18); # Peter Luschny, Aug 20 2014
MATHEMATICA
q[n_, k_] := 1;
r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}]
u[0, x_] := 1; u[n_, x_] := (x + n)*u[n - 1, x]
p[n_, k_] := Coefficient[u[n, x], x, k]
v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
Table[v[n], {n, 0, 18}] (* A193651 *)
TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
Table[r[k], {k, 0, 8}] (* 2^k *)
TableForm[Table[p[n, k], {n, 0, 6}, {k, 0, n}]] (* A130534 *)
Table[((2 n + 1)!! + 1)/2, {n, 0, 18}] (* or *)
Table[(2^n Gamma[n + 3/2])/Sqrt[Pi] + 1/2, {n, 0, 18}] (* or *)
Table[2^n Pochhammer[1/2, n + 1] + 1/2, {n, 0, 18}] (* Michael De Vlieger, Apr 25 2016 *)
PROG
(Sage)
def A():
n, a, b = 1, 1, 2
yield a
while True:
yield b
n += 1
a, b = b, ((2*(b-a)*n + a)*n - b)/(n-1)
A193651 = A()
[next(A193651) for i in range(19)] # Peter Luschny, Aug 20 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Aug 02 2011
EXTENSIONS
New name from Peter Luschny, Aug 20 2014
STATUS
approved