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A128195 Double Variations. 5

%I

%S 1,9,65,511,4743,52525,683657,10256775,174369527,3313030741,

%T 69573667065,1600194389599,40004859842375,1080131215965309,

%U 31323805263469097

%N Double Variations.

%C VarScheme(k,n) = (n*k+1)*(VarScheme(k,n-1) + k^n), VarScheme(k,0) = 1. a(n) is the third row of this scheme, a(n) = VarScheme(2,n).

%C k | n -> the array A126062:

%C [0]..1,..1,...1,.....1,......1,.......1,.........1,..........1,............1

%C [1]..1,..4,..15,....64,....325,....1956,.....13699,.....109600,.......986409

%C [2]..1,..9,..65,...511,...4743,...52525,....683657,...10256775,....174369527

%C [3]..1,.16,.175,..2020,..27313,..440896,...8390875,..184647364,...4616348125

%C [4]..1,.25,.369,..5629,.100045,.2122449,..53163625,.1542220261,..50895431301

%C [5]..1,.36,.671,.12736,.280581,.7376356,.229151411,.8252263296,.338358810761

%C The second row counts the variations of n distinct objects A007526.

%C The second column is sequence A000290. The third column is sequence A005917.

%H P. Luschny, <a href="http://www.luschny.de/math/seq/variations.html">Variants of Variations</a>.

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

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

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

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

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

%F Recursive form:

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

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

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

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

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

%F Generalized form: For x>0

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

%F Asymptotic formula:

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

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

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

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

%t a[0] = 1; a[n_] := a[n] = (2*n+1)*(a[n-1] + 2^n); Table[a[n], {n, 0, 14}] (* _Jean-Fran├žois Alcover_, Jul 29 2013 *)

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

%K easy,nonn

%O 0,2

%A _Peter Luschny_, Feb 26 2007

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Last modified May 10 04:08 EDT 2021. Contains 343748 sequences. (Running on oeis4.)