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A128196 A weighted sum of quotients of double factorials. 4
1, 3, 13, 73, 527, 4775, 52589, 683785, 10257031, 174370039, 3313031765, 69573669113, 1600194393695, 40004859850567, 1080131215981693 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n) is the sum of rows in the following triangle (A126063):

T(n,k) (n,k>=0)

...........1.

...........1,.......2

...........3,.......6,.......4.

..........15,......30,......20,.......8

.........105,.....210,.....140,......56,.....16

.........945,....1890,....1260,.....504,....144,....32

.......10395,...20790,...13860,....5544,...1584,...352,....64

......135135,..270270,..180180,...72072,..20592,..4576,...832,..128

First column is A001147, second column is A097801.

The diagonal is A000079, the subdiagonal is A014480.

Let H be the diagonal matrix diag(1,2,4,8,...) and

let G be the matrix (n!! defined as A001147(n), -1!! = 1):

(-1)!!/(-1)!!

1!!/(-1)!! 1!!/1!!

3!!/(-1)!! 3!!/1!! 3!!/3!!

5!!/(-1)!! 5!!/1!! 5!!/3!! 5!!/5!!

...

Then T = G*H. [Gottfried Helms]

LINKS

Table of n, a(n) for n=0..14.

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-Fran├žois Alcover, Jun 28 2013 *)

CROSSREFS

Cf. A128195, A001147, A126063.

Sequence in context: A059294 A124468 A205572 * A162161 A119013 A190878

Adjacent sequences:  A128193 A128194 A128195 * A128197 A128198 A128199

KEYWORD

easy,nonn

AUTHOR

Peter Luschny, Feb 26 2007

STATUS

approved

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Last modified December 21 05:22 EST 2014. Contains 252296 sequences.