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 A193651 a(n) = ((2*n + 1)!! + 1)/2. 4
 1, 2, 8, 53, 473, 5198, 67568, 1013513, 17229713, 327364538, 6874655288, 158117071613, 3952926790313, 106729023338438, 3095141676814688, 95949391981255313, 3166329935381425313, 110821547738349885938, 4100397266318945779688 (list; graph; refs; listen; history; text; internal format)
 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 Cf. A001147, A193649, A130534. Sequence in context: A145157 A323871 A183945 * A195979 A203109 A197795 Adjacent sequences:  A193648 A193649 A193650 * A193652 A193653 A193654 KEYWORD nonn,easy AUTHOR Clark Kimberling, Aug 02 2011 EXTENSIONS New name from Peter Luschny, Aug 20 2014 STATUS approved

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Last modified May 9 00:09 EDT 2021. Contains 343685 sequences. (Running on oeis4.)