A126062 -id:A126062 - cp's OEIS frontend

A128195 a(n) = (2n + 1)(a(n - 1) + 2^n) for n >= 1, a(0) = 1.

1, 9, 65, 511, 4743, 52525, 683657, 10256775, 174369527, 3313030741, 69573667065, 1600194389599, 40004859842375, 1080131215965309, 31323805263469097, 971037963168557815, 32044252784564570583, 1121548847459764557925, 41497307356011298342553, 1618394986884440655806799

Offset: 0

Peter Luschny, Feb 26 2007

P. Luschny, Variants of Variations.

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

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

a[0] = 1; a[n_] := a[n] = (2*n+1)*(a[n-1] + 2^n); Table[a[n], {n, 0, 14}] (* Jean-François Alcover, Jul 29 2013 *)

a(n) = A126062(2, n), double variations.
a(n) = (2n+1)!/(n! 2^n) Sum(k=0..n, 4^k*k!/(2k)!) [Gottfried Helms]
a(n) = 2^n (2n+1) Sum(k=0..n, Gamma(n+1/2)/Gamma(k+1/2))
a(n) = 2^(n+1) Gamma(n+3/2) Sum(k=0..n, 1/Gamma(k+1/2))
a(n) = A128196(n)*A005408(n)
a(n) = A128196(n+1)-A000079(n+1)
Recursive form:
a(n) = 2^(n+1)*v(n+1/2) with v(x) = if x <= 1 then x else x(v(x-1)+1).
a(n) = (2n+1)*(a(n-1)+2^n), a(0) = 1 [Wolfgang Thumser]
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+1)(x+1/2)(exp(1) Gamma(x+1/2,1) + K Gamma(x+1/2))
Asymptotic formula:
a(n) ~ 2^(n+5/2)*Gamma(n+3/2)
a(n) ~ (exp(1)+K)*2^(n+1)*(n+1/2)!
a(n) ~ M(2n+1)(2exp(-1)(n-1/(24*n+19/10*1/n)))^n

A128198 Array read by antidiagonals. A scheme of arrangements: ArrScheme(k,n) = VarScheme(k,n-1) + k^n; ArrScheme(k,0) = 1. VarScheme(k,n) = (nk+1)(VarScheme(k,n-1) + k^n); VarScheme(k,0) = 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 13, 16, 1, 1, 5, 25, 73, 65, 1, 1, 6, 41, 202, 527, 326, 1, 1, 7, 61, 433, 2101, 4775, 1957, 1

Offset: 0

Peter Luschny, Mar 02 2007

The second row (k=1) is sequence A000522 counting the arrangements of a set with n elements and the third row (k=2) is the sequence A128196. Cf. the scheme of variations A128195.

			Array begins:
[k=0] 1, 1, 1, 1, 1, 1, 1, 1
[k=1] 1, 2, 5, 16, 65, 326, 1957, 13700
[k=2] 1, 3, 13, 73, 527, 4775, 52589, 683785
[k=3] 1, 4, 25, 202, 2101, 27556, 441625, 8393062
[k=4] 1, 5, 41, 433, 5885, 101069, 2126545, 53180009
[k=5] 1, 6, 61, 796, 13361, 283706, 7391981, 229229536
[k=6] 1, 7, 85, 1321, 26395, 667651, 20743837, 767801905
[k=7] 1, 8, 113, 2038, 47237, 1386680, 50038129, 2152463090

P. Luschny, Variants of Variations.

Cf. A000522, A128195, A128196, A126062.

VarScheme := (k,n) -> `if`(n=0,1,(n*k+1)*(VarScheme(k,n-1)+k^n)); ArrScheme := (k,n) -> `if`(n=0,1, VarScheme(k,n-1)+k^n);

cp's OEIS Frontend

A128195 a(n) = (2n + 1)(a(n - 1) + 2^n) for n >= 1, a(0) = 1.

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A128198 Array read by antidiagonals. A scheme of arrangements: ArrScheme(k,n) = VarScheme(k,n-1) + k^n; ArrScheme(k,0) = 1. VarScheme(k,n) = (nk+1)(VarScheme(k,n-1) + k^n); VarScheme(k,0) = 1.

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cp's OEIS Frontend

A128195 a(n) = (2*n + 1)*(a(n - 1) + 2^n) for n >= 1, a(0) = 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A128198 Array read by antidiagonals. A scheme of arrangements: ArrScheme(k,n) = VarScheme(k,n-1) + k^n; ArrScheme(k,0) = 1. VarScheme(k,n) = (n*k+1)*(VarScheme(k,n-1) + k^n); VarScheme(k,0) = 1.

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Keywords

Comments

Examples

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Programs

Maple

A128195 a(n) = (2n + 1)(a(n - 1) + 2^n) for n >= 1, a(0) = 1.

A128198 Array read by antidiagonals. A scheme of arrangements: ArrScheme(k,n) = VarScheme(k,n-1) + k^n; ArrScheme(k,0) = 1. VarScheme(k,n) = (nk+1)(VarScheme(k,n-1) + k^n); VarScheme(k,0) = 1.