A060099 G.f.: 1/((1-x^2)^3*(1-x)^4).

1, 4, 13, 32, 71, 140, 259, 448, 742, 1176, 1806, 2688, 3906, 5544, 7722, 10560, 14223, 18876, 24739, 32032, 41041, 52052, 65429, 81536, 100828, 123760, 150892, 182784, 220116, 263568, 313956, 372096, 438957, 515508, 602889, 702240, 814891, 942172, 1085623

Offset: 0

Author

Wolfdieter Lang, Apr 06 2001

Keywords

• nonn
• easy

Comments

Fourth column (m=3) of triangle A060098.

Partial sums of A038163.

Equals the tetrahedral numbers, [1, 4, 10, 20, ...] convolved with the aerated triangular numbers, [1, 0, 3, 0, 6, 0, 10, ...]. [Gary W. Adamson, Jun 11 2009]

References

• B. Broer, Hilbert series for modules of covariants, in Algebraic Groups and Their Generalizations..., Proc. Sympos. Pure Math., 56 (1994), Part I, 321-331. See p. 329.

Links

• Peter J. C. Moses, Table of n, a(n) for n = 0..9999
• Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 20.
• Index entries for linear recurrences with constant coefficients, signature (4,-3,-8,14,0,-14,8,3,-4,1).

Crossrefs

Cf. A002620, A002624, A096338.

Cf. A001752 (for the similar series 1/((1-x)^4*(1-x^2))).

Cf. A028346 (for the similar series 1/((1-x)^4*(1-x^2)^2)).

Programs

Mathematica

a[n_]:=If[OddQ[n],((1+n) (3+n) (5+n)^2 (7+n) (9+n))/5760,((2+n) (4+n) (6+n) (8+n) (15+10 n+n^2))/5760]; Map[a,Range[0,100]] (* Peter J. C. Moses, Mar 24 2013 *)
CoefficientList[Series[1/((1-x^2)^3*(1-x)^4),{x,0,100}],x] (* Peter J. C. Moses, Mar 24 2013 *)
LinearRecurrence[{4,-3,-8,14,0,-14,8,3,-4,1},{1,4,13,32,71,140,259,448,742,1176},40] (* Harvey P. Dale, Apr 06 2018 *)

Formula

a(n) = Sum_{} A060098(n+3, 3).

G.f.: 1/((1-x)^7*(1+x)^3).