A122576 - OEIS

%I #54 Nov 03 2024 09:34:01

%S -1,3,-12,20,-45,63,-112,144,-225,275,-396,468,-637,735,-960,1088,

%T -1377,1539,-1900,2100,-2541,2783,-3312,3600,-4225,4563,-5292,5684,

%U -6525,6975,-7936,8448,-9537,10115,-11340,11988,-13357,14079,-15600,16400,-18081,18963,-20812,21780,-23805

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

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

%C Unsigned = partial sums of positive integers of A129194. - _Omar E. Pol_, Aug 22 2011

%C Unsigned, see A212760. - _Clark Kimberling_, May 29 2012

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

%H Vincenzo Librandi, <a href="/A122576/b122576.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (-1,3,3,-3,-3,1,1).

%F a(n) = n*(n+1)/8 * ((2*n+1)*(-1)^n - 1). - _Ralf Stephan_, Jan 01 2014

%F 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

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

%p 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

%t Table[(n + 1) (n + 2) (2 n + 3 + (-1)^n) (-1)^(n + 1)/8, {n, 0, 50}] (* _Wesley Ivan Hurt_, Jul 22 2014 *)

%t 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 *)

%o (Magma) [(n+1)*(n+2)*(2*n+3+(-1)^n)*(-1)^(n+1)/8 : n in [0..50]]; // _Wesley Ivan Hurt_, Jul 22 2014

%o (SageMath)

%o def A122576(n): return n*(n+1)*((2*n+1)*(-1)^n - 1)//8

%o [A122576(n) for n in range(1,51)] # _G. C. Greubel_, Nov 02 2024

%Y Cf. A026741, A098023, A143120, A212760.

%K sign,easy

%O 1,2

%A _Roger L. Bagula_, Sep 17 2006

%E 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.