A157371 a(n) = (n+1)*(9-9*n+5*n^2-n^3).

9, 8, 9, 0, -55, -216, -567, -1216, -2295, -3960, -6391, -9792, -14391, -20440, -28215, -38016, -50167, -65016, -82935, -104320, -129591, -159192, -193591, -233280, -278775, -330616, -389367, -455616, -529975, -613080, -705591, -808192, -921591, -1046520, -1183735, -1334016, -1498167

Offset: 0

Author

Paul Curtz, Feb 28 2009

Keywords

• sign
• easy
• changed

Comments

This is the fourth in a family of sequences that appear in columns on pages 36 and 56 of the reference: (i) sequence n+1, A000029, (ii) sequence (n+1)*(1-n), A147998 and (iii) (n+1)*(5-5*n+2*n^2), A152064.

First differences along columns shown on page 56 of the reference are columns of what is shown on page 36 of the reference. Example: the third column of page 56, A152064, has first differences which constitute the third column p page 36, A140811.

References

• Paul Curtz, Intégration numérique des systèmes différentiels à conditions initiales, Centre de Calcul Scientifique de l'Armement, Note 12, Arcueil (1969).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Programs

Magma

[(n+1)*(9-9*n+5*n^2-n^3): n in [0..40] ]; // Vincenzo Librandi, Jul 14 2011

Mathematica

LinearRecurrence[{5,-10,10,-5,1},{9,8,9,0,-55},40] (* or *) Table[(n+1)(9-9n+5n^2-n^3),{n,0,40}] (* or *) CoefficientList[ Series[ (55x^3- 59x^2+ 37x-9)/ (x-1)^5,{x,0,40}],x] (* Harvey P. Dale, Jul 13 2011 *)

PARI

a(n)=(n+1)*(9-9*n+5*n^2-n^3) \\ Charles R Greathouse IV, Oct 16 2015

Formula

First differences: a(n+1)-a(n) = -A141530(n).

Fourth differences: a(n+4)-4*a(n+3)+6*a(n+2)-4*a(n+1)+a(n) = -24 = -A010863(n).

From Harvey P. Dale, Jul 13 2011: (Start)

a(0)=9, a(1)=8, a(2)=9, a(3)=0, a(4)=-55, a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5).

G.f.: (9-37*x+59*x^2-55*x^3)/(1-x)^5. (End)

E.g.f.: (9 - x + x^2 - 2*x^3 - x^4)*exp(x). - G. C. Greubel, Feb 02 2018

Extensions

Edited, extended by R. J. Mathar, Sep 25 2009