|
|
A010844
|
|
a(n) = 2*n*a(n-1) + 1 with a(0) = 1.
|
|
34
|
|
|
1, 3, 13, 79, 633, 6331, 75973, 1063623, 17017969, 306323443, 6126468861, 134782314943, 3234775558633, 84104164524459, 2354916606684853, 70647498200545591, 2260719942417458913, 76864478042193603043, 2767121209518969709549, 105150605961720848962863
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
Related to Incomplete Gamma Function at 1/2. - Michael Somos, Mar 26 1999
For positive n, a(n) is equal to 2^n times the permanent of the n X n matrix with 3/2's along the main diagonal, and 1's everywhere else. - John M. Campbell, Jul 09 2011
Number of ways to sort a spreadsheet with n columns. (A subset of columns is chosen to sort on. These columns are ordered from major to minor, and each designated as to whether to sort by ascending or descending order. For example a spreadsheet with columns A,B,C,D could be sorted by column D ascending, then by column B descending, or any of 632 other ways.) - Marc LeBrun, Dec 07 2013
a(n) is a specific instance of sequences having the form b(0) = x, b(n) = a*n*b(n-1) + k for n >= 1. (Here x = 1, a = 2, and k = 1). Sequences of this form have a closed form of b(n) = n!*a^n*x + k*Sum_{j=1..n} n!*a^(n-j)/j!. - Gary Detlefs, Mar 26 2018
|
|
REFERENCES
|
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 262.
|
|
LINKS
|
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
|
|
FORMULA
|
a(n) = floor(n! * e^(1/2) * 2^n) = n! * Sum_{k=0..n} 2^(n-k) / k! (i.e. binomial transform of (2n)!! = n!*2^n) = n! * (e^(1/2) * 2^n - Sum_{k >= n+1} 2^(n-k) / k!)). - Michael Somos, Mar 26 1999
Sum_{n >= 1} 1/a(n) = 0.4246665348160769533082551230... - Cino Hilliard, Aug 19 2003
a(n) = Sum_{k=0..n} P(n, k)*2^k, where P(n,k) = n!/(n-k)!. - Ross La Haye, Aug 29 2005
G.f.: 1/(1 - x - 2*x/(1 - 2*x/(1 - x - 4*x/(1 - 4*x/(1 - x - 6*x/(1 - 6*x/(1 - x - 8*x/(1 - 8*x/(1 - x - 10*x/(1 - ... (continued fraction).
G.f.: 1/Q(0), where Q(k) = 1 - x*(4*k+3) - 4*x^2*(k+1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Sep 30 2013
a(n) = Sum_{k=0..n} C(n,k)*k!*2^k. - Marc LeBrun, Dec 07 2013
0 = a(n)*(2*a(n+1) - 5*a(n+2) + a(n+3)) + a(n+1)*(a(n+1) + a(n+2) - a(n+3)) + a(n+2)*a(n+2) if n > -2. - Michael Somos, Jan 02 2014
a(n) + (-2*n-1)*a(n-1) + 2*(n-1)*a(n-2) = 0. - R. J. Mathar, Jan 31 2014
a(n) = hypergeometric_U(1, n+2, 1/2)/2. - Peter Luschny, Nov 26 2014
a(n) = Integral_{x >= 0} (2*x + 1)^n*exp(-x) dx. (Cf. A000354.)
The e.g.f. y = exp(x)/(1 - 2*x) satisfies the differential equation (1 - 2*x)*y' = (3 - 2*x)*y. R. J. Mathar's recurrence above follows easily from this.
The sequence b(n) := 2^n*n! also satisfies R. J. Mathar's recurrence with b(0) = 1 and b(1) = 2. This leads to the continued fraction representation a(n) = 2^n*n!*( 1 + 1/(2 - 2/(5 - 4/(7 - ... - (2*n - 2)/(2*n + 1) )))) for n >= 2. Taking the limit gives the continued fraction representation exp(1/2) = 1 + 1/(2 - 2/(5 - 4/(7 - ... - (2*n - 2)/((2*n + 1) - ... )))). (End)
a(n) = n!*2^n*hypergeom([-n], [-n], 1/2). - Peter Luschny, Jul 28 2024
|
|
EXAMPLE
|
a(3) = 2*3*a(2) + 1 = 6*13 + 1 = 79.
G.f. = 1 + 3*x + 13*x^2 + 79*x^3 + 633*x^4 + 6331*x^5 + 75973*x^6 + 1063623*x^7 + ...
|
|
MAPLE
|
G:=(x, a, k, n)-> n!*a^n*x + k*sum(n!*a^(n-j)/j!, j=1..n); seq(G(1, 2, 1, n), n = 0..20) # Gary Detlefs, Mar 26 2018
a := n -> 2^n*add((n!/k!)*(1/2)^k, k=0..n):
seq(simplify(2^n*KummerU(-n, -n, 1/2)), n = 0..19); # Peter Luschny, May 10 2022
|
|
MATHEMATICA
|
Table[ Gamma[ n, 1/2 ]*Exp[ 1/2 ]*2^(n-1), {n, 1, 24} ]
a[ n_] := If[ n<0, 0, Floor[ n! E^(1/2) 2^n ]] (* Michael Somos, Sep 04 2013 *)
nxt[{n_, a_}]:={n+1, 2*a(n+1)+1}; NestList[nxt, {0, 1}, 20][[All, 2]] (* Harvey P. Dale, Jan 06 2022 *)
a[n_] := n! 2^n Hypergeometric1F1[-n, -n, 1/2];
|
|
PROG
|
(PARI) {a(n) = if( n<0, 0, n! * sum(k=0, n, 2^(n-k) / k!))} /* Michael Somos, Sep 04 2013 */
|
|
CROSSREFS
|
|
|
KEYWORD
|
easy,nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|