A161128 a(n) = n!*(1/1 + 1/2 + ... + 1/n) - (1! + 2! + ... + n!).

0, 0, 0, 2, 17, 121, 891, 7155, 63351, 617463, 6590727, 76589127, 963486567, 13052781927, 189537379047, 2937560365287, 48409889869287, 845393769958887, 15596602532173287, 303139660882458087, 6191620542649779687

Offset: 0

Author

Emeric Deutsch, Jul 14 2009

Keywords

• nonn

Comments

a(n) is the number of cycles that cannot be written in the form (j,j+1,j+2,...), in all permutations of {1,2,...,n}. Example: a(3)=2 because in (1)(2)(3), (1)(23), (12)(3), (13)(2), (123), (132) we have 0+0+0+1+0+1 = 2 such cycles.

Links

• Robert Israel, Table of n, a(n) for n = 0..449

Crossrefs

Cf. A000254, A007489.

Programs

Magma

[0] cat [Factorial(n)*HarmonicNumber(n) - (&+[Factorial(k): k in [1..n]]): n in [1..30]]; // G. C. Greubel, Oct 14 2018

Maple

a := proc (n) options operator, arrow: factorial(n)*harmonic(n)-add(factorial(j), j = 1 .. n) end proc: seq(a(n), n = 0 .. 22);
# Alternative:
f:= gfun:-rectoproc({(n+1)*n*a(n+3) - n*(2*n^2+8*n+7)*a(n+2) + (n+2)*(n^3+5*n^2+6*n+1)*a(n+1) - (n+1)^3*(n+2)*a(n), a(0)=0, a(1)=0, a(2)=0, a(3)=2},a(n),remember):
map(f, [$0..30]); # Robert Israel, Apr 11 2018

Mathematica

Table[n!*HarmonicNumber[n] - Sum[k!, {k,1,n}], {n,0,30}] (* G. C. Greubel, Oct 14 2018 *)

PARI

a(n) = n!*sum(k=1, n, 1/k) - sum(k=1, n, k!); \\ Michel Marcus, Apr 11 2018

Formula

a(n) = A000254(n) - A007489(n).

(n+1)*n*a(n+3) - n*(2*n^2+8*n+7)*a(n+2) + (n+2)*(n^3+5*n^2+6*n+1)*a(n+1) - (n+1)^3*(n+2)*a(n) = 0. - Robert Israel, Apr 11 2018