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COMMENT
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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]
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FORMULA
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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
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