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A024167 a(n) = n!*(1 - 1/2 + 1/3 - ... + c/n), where c = (-1)^(n+1). 17
1, 1, 5, 14, 94, 444, 3828, 25584, 270576, 2342880, 29400480, 312888960, 4546558080, 57424792320, 948550176000, 13869128448000, 256697973504000, 4264876094976000, 87435019510272000, 1627055289796608000 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Stirling transform of (-1)^n*a(n-1) = [0, 1, -1, 5, -14, 94, ...] is A000629(n-2) = [0, 1, 2, 6, 26, ...]. - Michael Somos, Mar 04 2004

Stirling transform of a(n) = [1, 1, 5, 14, 94, ...] is A052882(n) =[1, 2, 9, 52, 375, ...]. - Michael Somos, Mar 04 2004

a(n) is the number of n-permutations that have a cycle with length greater than n/2. - Geoffrey Critzer, May 28 2009

From Jens Voß, May 07 2010: (Start)

A024167(4n) is divisible by 6n+1 for all n >= 1; the quotient of A024167(4n) and 6n+1 is A177188(n).

A024167(4n+3) is divisible by 6n+5 for all n >= 0; the quotient of A024167(4n+3) and 6n+5 is A177174(n). (End)

LINKS

Table of n, a(n) for n=1..20.

FORMULA

E.g.f.: log(1+x)/(1-x). - Vladeta Jovovic, Aug 25 2002

a(n) = a(n-1) + a(n-2) * (n-1)^2, n>1. - Michael Somos, Oct 29, 2002

b(n) = n! satisfies the above recurrence with b(1) = 1, b(2) = 2. This gives the finite continued fraction expansion a(n)/n! = 1/(1+1^2/(1+2^2/(1+3^2/(1+...+(n-1)^2/1)))). Cf. A142979. - Peter Bala, Jul 17 2008

a(n) = A081358(n) - A092691(n). - Gary Detlefs, Jul 09 2010

E.g.f.: x/(x-1)/G(0) where G(k)= -1 + (x-1)*k + x*(k+1)^2/G(k+1); (continued fraction, Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Aug 18 2012

a(n) ~ log(2)*n!. - Daniel Suteu, Dec 03 2016

a(n) = 1/2*n!*((-1)^n*(digamma((n+1)/2) - digamma((n+2)/2)) + log(4)). - Daniel Suteu, Dec 03 2016

EXAMPLE

G.f. = x + x^2 + 5*x^3 + 14*x^4 + 94*x^5 + 444*x^6 + 3828*x^7 + 25584*x^8 + ...

MATHEMATICA

f[k_] := k (-1)^(k + 1)

t[n_] := Table[f[k], {k, 1, n}]

a[n_] := SymmetricPolynomial[n - 1, t[n]]

Table[a[n], {n, 1, 18}]    (* A024167 signed *)

(* Clark Kimberling, Dec 30 2011 *)

a[ n_] := If[ n < 0, 0, n! Sum[ -(-1)^k / k, {k, n}]]; (* Michael Somos, Nov 28 2013 *)

a[ n_] := If[ n < 0, 0, n! (PolyGamma[n + 1] - PolyGamma[(n + Mod[n, 2, 1]) / 2])]; (* Michael Somos, Nov 28 2013 *)

a[ n_] := If[ n < 1, 0, (-1)^Quotient[n, 2] SymmetricPolynomial[ n - 1, Table[ -(-1)^k k, {k, n}]]]; (* Michael Somos, Nov 28 2013 *)

PROG

(PARI) {a(n) = if( n<0, 0, n! * polcoeff( log(1 + x + x * O(x^n)) / (1 - x), n))}; /* Michael Somos, Mar 02 2004 */

CROSSREFS

Cf. A000254.

Cf. A142979.

Cf. A177174, A177188. - Jens Voß, May 07 2010

Sequence in context: A224245 A238380 A183307 * A077262 A184439 A267351

Adjacent sequences:  A024164 A024165 A024166 * A024168 A024169 A024170

KEYWORD

nonn,easy,changed

AUTHOR

Clark Kimberling

EXTENSIONS

More terms from Benoit Cloitre, Jan 27 2002

STATUS

approved

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Last modified December 7 22:57 EST 2016. Contains 278899 sequences.