A158802 a(n) = n * n! * b(n), where b(n) = ((n-1)*(n-3)*b(n-1) - b(n-2) + b(n-3))/(n*(n - 1)) and b(0) = b(1) = 1, b(2) = -1.

0, 1, -4, 0, 16, 10, 12, 182, 1120, 7452, 58640, 520784, 5142144, 55929640, 664505744, 8562670920, 118939979008, 1771631324848, 28168269788160, 476151820931168, 8526830353553920, 161255217263900256

Offset

0,3

References

Martin Braun, 'Differential Equations and Their Applications: An Introduction to Applied Mathematics', Texts in Applied Mathematics, Vol. 11, Springer, 4th ed., 1992, pp. 194-196, Example 5.

Links

G. C. Greubel, Table of n, a(n) for n = 0..445

Formula

a(n) = n * n! * b(n), where b(n) = ((n-1)*(n-3)*b(n-1) - b(n-2) + b(n-3))/(n*(n - 1)) and b(0) = b(1) = 1, b(2) = -1.

From G. C. Greubel, Dec 07 2021: (Start)

a(n) = (n/((n-1)*(n-2)*(n-3)))*((n-2)*(n-3)^2*a(n-1) - (n-1)*(n-3)*a(n-2) + (n-1)*(n-2)^2*a(n-3)), with a(0) = 0, a(1) = 1, a(2) = -4, a(3) = 0.

a(n) = n*n!*b(n), where y(t) = Sum_{n>=0} b(n)*t^n satisfies the differential equation (1-t)*y''(t) + y'(t) + (1-t)*y(t) = 0.

a(n) = n*n!*b(n), where y(t) = Sum_{n>=0} b(n)*t^n = (((1-t)*((BesselJ[0, 1] + 4*BesselJ[1, 1] - BesselJ[2, 1])*BesselY[1, t-1] - BesselJ[1, 1-t]*(BesselY[0, -1] - 4*BesselY[1, -1] - BesselY[2, -1])))/((BesselJ[0, 1] - BesselJ[2, 1]) BesselY[1, -1] + BesselJ[1, 1] (BesselY[0, -1] - BesselY[2, -1]))). (End)

Mathematica

b[0]:=1; b[1]:=1; b[2]:=-1;
b[n_]:= b[n]= ((n-1)*(n-3)*b[n-1] - b[n-2] + b[n-3])/(n*(n-1));
Table[n*n!*b[n], {n, 0, 30}]

Prog

(Sage)
@CachedFunction
def a(n):
    if (n<3): return (-1)^(n+1)*n^2
    elif (n==3): return 0
    else: return (n/((n-1)*(n-2)*(n-3)))*((n-2)*(n-3)^2*a(n-1) -(n-1)*(n-3)*a(n-2) +(n-1)*(n-2)^2*a(n-3))
[a(n) for n in (0..30)] # G. C. Greubel, Dec 07 2021

Crossrefs

Sequence in context: A167314 A208451 A207541 * A230280 A030212 A167359

Adjacent sequences: A158799 A158800 A158801 * A158803 A158804 A158805

Keyword

sign

Author

Roger L. Bagula, Mar 27 2009

Extensions

Edited by G. C. Greubel, Dec 07 2021

Status

approved