A122576 Expansion of g.f.: -x*(1 - 2*x + 6*x^2 - 2*x^3 + x^4)/((1-x)^3*(1+x)^4).

-1, 3, -12, 20, -45, 63, -112, 144, -225, 275, -396, 468, -637, 735, -960, 1088, -1377, 1539, -1900, 2100, -2541, 2783, -3312, 3600, -4225, 4563, -5292, 5684, -6525, 6975, -7936, 8448, -9537, 10115, -11340, 11988, -13357, 14079, -15600, 16400, -18081, 18963, -20812, 21780, -23805

Offset: 1

Author

Roger L. Bagula, Sep 17 2006

Keywords

• sign
• easy

Comments

Unsigned = row sums of triangle A143120 and Sum_{n>=1} n*A026741(n). - Gary W. Adamson, Jul 26 2008

Unsigned = partial sums of positive integers of A129194. - Omar E. Pol, Aug 22 2011

Unsigned, see A212760. - Clark Kimberling, May 29 2012

References

• Roger G. Newton, Scattering Theory of Waves and Particles, McGraw Hill, 1966; p. 254.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (-1,3,3,-3,-3,1,1).

Crossrefs

Cf. A026741, A098023, A143120, A212760.

Programs

Magma

[(n+1)*(n+2)*(2*n+3+(-1)^n)*(-1)^(n+1)/8 : n in [0..50]]; // Wesley Ivan Hurt, Jul 22 2014

Maple

A122576:=n->(n+1)*(n+2)*(2*n+3+(-1)^n)*(-1)^(n+1)/8: seq(A122576(n), n=0..50); # Wesley Ivan Hurt, Jul 22 2014

Mathematica

Table[(n + 1) (n + 2) (2 n + 3 + (-1)^n) (-1)^(n + 1)/8, {n, 0, 50}] (* Wesley Ivan Hurt, Jul 22 2014 *)
CoefficientList[Series[(1 -2 x +6 x^2 -2 x^3 +x^4)/((x-1)^3 (x+1)^4), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 23 2014 *)

SageMath

def A122576(n): return n*(n+1)*((2*n+1)*(-1)^n - 1)//8
[A122576(n) for n in range(1,51)] # G. C. Greubel, Nov 02 2024

Formula

a(n) = n*(n+1)/8 * ((2*n+1)*(-1)^n - 1). - Ralf Stephan, Jan 01 2014

a(n) = (n+1)*(n+2)*(2*n+3+(-1)^n)*(-1)^(n+1)/8 (with offset of 0). - Wesley Ivan Hurt, Jul 22 2014

E.g.f.: -(1/8)*x*( (6 - 9*x + 2*x^2)*exp(-x) + (2+x)*exp(x) ). - G. C. Greubel, Nov 02 2024

Extensions

Edited by N. J. A. Sloane, May 20 2007. The simple generating function now used to define the sequence was found by an anonymous correspondent.