A186638 a(0)=a(1)=a(2)=0; thereafter a(n) = n*a(n-1) + n*a(n-2)/(n-2) + (-1)^(n-1)*4/(n-2).

0, 0, 0, 4, 14, 78, 488, 3526, 28858, 264256, 2678632, 29787932, 360669542, 4723907966, 66555492656, 1003783052878, 16136592266226, 275459689319104, 4976428074043376, 94860000118416084, 1902729366895036542, 40062161968084543054, 883460565601444487384, 20363470614798268185558, 489687069917632739530538, 12264310955130816605856448

Offset: 0

Author

N. J. A. Sloane, Feb 24 2011

Keywords

• nonn

Comments

Muir gives this recurrence without specifying the initial values.

In general, for the same recurrence a(n) = n*a(n-1) + n*a(n-2)/(n-2) + (-1)^(n-1)*4/(n-2), with a(1)=0, a(2)=0, a(3)=m, is a(n) ~ c * n!, where c = exp(-2) + (BesselI(0,2)-BesselI(1,2))*(m-1)/3. Set m=4 for this sequence and m=1 for A000179. - Vaclav Kotesovec, May 05 2015

References

• T. Muir, A Treatise on the Theory of Determinants. Dover, NY, 1960, Sect. 132, p. 112.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..69

Crossrefs

A000179 satisfies essentially the same recurrence.

Programs

Maple

W:=proc(n) option remember; if n <= 2 then 0 else
n*W(n-1)+n*W(n-2)/(n-2)+(-1)^(n-1)*4/(n-2); fi; end;

Mathematica

Flatten[{0,0,RecurrenceTable[{a[2]==0,a[3]==4,a[n]==n*a[n-1]+n*a[n-2]/(n-2)+(-1)^(n-1)*4/(n-2)},a,{n,2,20}]}] (* Vaclav Kotesovec, May 05 2015 *)

Maxima

a[0]:0$ a[1]:0$ a[2]:0$ a[n]:=n*a[n-1]+n*a[n-2]/(n-2)+4*(-1)^(n-1)/(n-2)$ makelist(a[n], n, 0, 25); /* Bruno Berselli, May 23 2011 */

Formula

Recurrence (for n>2): (n-2)*a(n) = (n^2 - 3*n + 3)*a(n-1) + (n^2 - 3*n + 3)*a(n-2) + (n-1)*a(n-3). - Vaclav Kotesovec, May 05 2015

a(n) ~ c * n!, where c = exp(-2) + BesselI(0,2) - BesselI(1,2) = 0.8242837309353508959489495107843515087389944891994982884067... . - Vaclav Kotesovec, May 05 2015