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

1, 2, 8, 53, 473, 5198, 67568, 1013513, 17229713, 327364538, 6874655288, 158117071613, 3952926790313, 106729023338438, 3095141676814688, 95949391981255313, 3166329935381425313, 110821547738349885938, 4100397266318945779688, 159915493386438885407813

Offset: 0

Author

Clark Kimberling, Aug 02 2011

Keywords

• nonn
• easy

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

Crossrefs

Cf. A001147, A193649, A130534.

Programs

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 *)

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

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

Extensions

New name from Peter Luschny, Aug 20 2014