Search: id:A128195
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%I A128195
%S A128195 1,9,65,511,4743,52525,683657,10256775,174369527,3313030741,69573667065,
%T A128195 1600194389599,40004859842375,1080131215965309,31323805263469097
%N A128195 Double Variations.
%C A128195 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 A128195 k | n -> the array A126062:
%C A128195 [0]..1,..1,...1,.....1,......1,.......1,.........1,..........1,............1
%C A128195 [1]..1,..4,..15,....64,....325,....1956,.....13699,.....109600,.......986409
%C A128195 [2]..1,..9,..65,...511,...4743,...52525,....683657,...10256775,....174369527
%C A128195 [3]..1,.16,.175,..2020,..27313,..440896,...8390875,..184647364,...4616348125
%C A128195 [4]..1,.25,.369,..5629,.100045,.2122449,..53163625,.1542220261,..50895431301
%C A128195 [5]..1,.36,.671,.12736,.280581,.7376356,.229151411,.8252263296,.338358810761
%C A128195 The second row counts the variations of n distinct objects A007526.
%C A128195 The second column is sequence A000290. The third column is sequence A005917.
%H A128195 P. Luschny, <a href="http://www.luschny.de/math/seq/variations.html">
               Variants of Variations</a>.
%F A128195 a(n) = (2n+1)!/(n! 2^n) Sum(k=0..n, 4^k*k!/(2k)!) [Gottfried Helms]
%F A128195 a(n) = 2^n (2n+1) Sum(k=0..n, Gamma(n+1/2)/Gamma(k+1/2))
%F A128195 a(n) = 2^(n+1) Gamma(n+3/2) Sum(k=0..n, 1/Gamma(k+1/2))
%F A128195 a(n) = A128196(n)*A005408(n)
%F A128195 a(n) = A128196(n+1)-A000079(n+1)
%F A128195 Recursive form:
%F A128195 a(n) = 2^(n+1)*v(n+1/2) with v(x) = if x <= 1 then x else x(v(x-1)+1).
%F A128195 a(n) = (2n+1)*(a(n-1)+2^n), a(0) = 1 [Wolfgang Thumser]
%F A128195 Note: The following constants will be used in the next formulas.
%F A128195 K = (1-exp(1)*Gamma(1/2,1))/Gamma(1/2)
%F A128195 M = sqrt(2)(1+exp(1)(Gamma(1/2)-Gamma(1/2,1)))
%F A128195 Generalized form: For x>0
%F A128195 a(x) = 2^(x+1)(x+1/2)(exp(1) Gamma(x+1/2,1) + K Gamma(x+1/2))
%F A128195 Asymptotic formula:
%F A128195 a(n) ~ 2^(n+5/2)*Gamma(n+3/2)
%F A128195 a(n) ~ (exp(1)+K)*2^(n+1)*(n+1/2)!
%F A128195 a(n) ~ M(2n+1)(2exp(-1)(n-1/(24*n+19/10*1/n)))^n
%p A128195 a := n -> `if`(n=0,1,(2*n+1)*(a(n-1)+2^n));
%Y A128195 Cf. A007526 (The number of variations), A128196 (A weighted sum of double 
               factorials), A126062.
%Y A128195 Sequence in context: A036731 A020234 A154996 this_sequence A103459 A100311 
               A120286
%Y A128195 Adjacent sequences: A128192 A128193 A128194 this_sequence A128196 A128197 
               A128198
%K A128195 easy,nonn
%O A128195 0,2
%A A128195 Peter Luschny (peter(AT)luschny.de), Feb 26 2007

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