A128195 a(n) = (2*n + 1)*(a(n - 1) + 2^n) for n >= 1, a(0) = 1.

1, 9, 65, 511, 4743, 52525, 683657, 10256775, 174369527, 3313030741, 69573667065, 1600194389599, 40004859842375, 1080131215965309, 31323805263469097, 971037963168557815, 32044252784564570583, 1121548847459764557925, 41497307356011298342553, 1618394986884440655806799

Offset

0,2

Links

Table of n, a(n) for n=0..19.

P. Luschny, Variants of Variations.

Formula

a(n) = A126062(2, n), double variations.

a(n) = (2n+1)!/(n! 2^n) Sum(k=0..n, 4^k*k!/(2k)!) [Gottfried Helms]

a(n) = 2^n (2n+1) Sum(k=0..n, Gamma(n+1/2)/Gamma(k+1/2))

a(n) = 2^(n+1) Gamma(n+3/2) Sum(k=0..n, 1/Gamma(k+1/2))

a(n) = A128196(n)*A005408(n)

a(n) = A128196(n+1)-A000079(n+1)

Recursive form:

a(n) = 2^(n+1)*v(n+1/2) with v(x) = if x <= 1 then x else x(v(x-1)+1).

a(n) = (2n+1)*(a(n-1)+2^n), a(0) = 1 [Wolfgang Thumser]

Note: The following constants will be used in the next formulas.

K = (1-exp(1)*Gamma(1/2,1))/Gamma(1/2)

M = sqrt(2)(1+exp(1)(Gamma(1/2)-Gamma(1/2,1)))

Generalized form: For x>0

a(x) = 2^(x+1)(x+1/2)(exp(1) Gamma(x+1/2,1) + K Gamma(x+1/2))

Asymptotic formula:

a(n) ~ 2^(n+5/2)*Gamma(n+3/2)

a(n) ~ (exp(1)+K)*2^(n+1)*(n+1/2)!

a(n) ~ M(2n+1)(2exp(-1)(n-1/(24*n+19/10*1/n)))^n

Maple

a := n -> `if`(n=0, 1, (2*n+1)*(a(n-1)+2^n));

Mathematica

a[0] = 1; a[n_] := a[n] = (2*n+1)*(a[n-1] + 2^n); Table[a[n], {n, 0, 14}] (* Jean-François Alcover, Jul 29 2013 *)

Crossrefs

Cf. A007526 (The number of variations), A128196 (A weighted sum of double factorials), A126062.

Sequence in context: A036731 A020234 A154996 * A103459 A339688 A100311

Adjacent sequences: A128192 A128193 A128194 * A128196 A128197 A128198

Keyword

easy,nonn

Author

Peter Luschny, Feb 26 2007

Status

approved