A128196 a(n) = (2*n - 1)*a(n - 1) + 2^n for n >= 1, a(0) = 1.

1, 3, 13, 73, 527, 4775, 52589, 683785, 10257031, 174370039, 3313031765, 69573669113, 1600194393695, 40004859850567, 1080131215981693, 31323805263501865, 971037963168623351, 32044252784564701655, 1121548847459764820069, 41497307356011298866841, 1618394986884440656855375

Offset

0,2

Comments

A weighted sum of quotients of double factorials.

a(n) are the row sum of triangle A126063.

Links

Table of n, a(n) for n=0..20.

P. Luschny, Variants of Variations.

Formula

a(n) = (2n)!/(n! 2^n) Sum(k=0..n, 4^k k!/(2k)!)

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

a(n) = Sum(k=0..n, 2^k n!!/k!!) [n!! defined as A001147(n), Gottfried Helms]

a(n) = Sum(k=0..n, 2^(2k-n)((n+1)! Catalan(n))/((k+1)! Catalan(k))) [Catalan(n) A000108]

a(n) = Sum(k=0..n, 2^(2k-n) QuadFact(n)/QuadFact(k)) [QuadFact(n) A001813]

a(n) = Sum(k=0..n, 2^(2k-n) (-1)^(n-k) A097388(n)/A097388(k) )

a(n) = A001147(n) Sum(k=0..n, 2^k / A001147(k))

a(n) = A128195(n)/A005408(n)

a(n) = A128195(n-1)+A000079(n) (if n>0)

Recursive form: a(n) = (2n-1)*a(n-1) + 2^n; a(0) = 1 [Gottfried Helms]

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(exp(1)*Gamma(x+1/2,1) + K*Gamma(x+1/2))

Asymptotic formula:

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

a(n) ~ M(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[n_] := Sum[2^k*((2*n-1)!!/(2*k-1)!!), {k, 0, n}]; Table[a[n], {n, 0, 14}] (* Jean-François Alcover, Jun 28 2013 *)

Crossrefs

Cf. A128195, A001147, A126063.

Sequence in context: A124468 A352302 A205572 * A367747 A162161 A119013

Adjacent sequences: A128193 A128194 A128195 * A128197 A128198 A128199

Keyword

easy,nonn

Author

Peter Luschny, Feb 26 2007

Status

approved