A157411 a(n) = 30*n^4 - 120*n^3 + 120*n^2 - 19.

-19, 11, -19, 251, 1901, 6731, 17261, 36731, 69101, 119051, 191981, 294011, 431981, 613451, 846701, 1140731, 1505261, 1950731, 2488301, 3129851, 3887981, 4776011, 5807981, 6998651, 8363501, 9918731, 11681261, 13668731, 15899501, 18392651

Offset: 0

Author

Paul Curtz, Feb 28 2009

Keywords

• sign
• easy

Comments

These are the numerators in column j=4 of the array in A140825 (reference p. 36).

The other columns in A140825 are represented by A000012, A005408, A140811 and A141530.

The link between these columns is given by the first differences: a(n+1) - a(n) = 30*A141530(n), where 30 = A027760(4) = A027760(3) = A027642(4) = A002445(2), then for j=3, A141530(n+1) - A141530(n) = A140070(2)*A140811(n).

References

• P. Curtz, Integration numerique des systemes differentiels a 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

[30*n^4 - 120*n^3 + 120*n^2 - 19: n in [0..50]]; // Vincenzo Librandi, Aug 07 2011

Mathematica

Table[30n^4-120n^3+120n^2-19,{n,0,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{-19,11,-19,251,1901},40] (* Harvey P. Dale, Mar 08 2015 *)

PARI

a(n)=30*n^4-120*n^3+120*n^2-19 \\ Charles R Greathouse IV, Oct 16 2015

Formula

a(n)= 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).

G.f.: (-19 + 106*x - 264*x^2 + 646*x^3 + 251*x^4)/(1-x)^5.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + 720. Fourth differences are constant, 720.

Extensions

Edited, one index corrected and extended by R. J. Mathar, Sep 17 2009