login
G.f.: 1/((1-x^2)^3*(1-x)^4).
8

%I #35 May 05 2023 10:22:08

%S 1,4,13,32,71,140,259,448,742,1176,1806,2688,3906,5544,7722,10560,

%T 14223,18876,24739,32032,41041,52052,65429,81536,100828,123760,150892,

%U 182784,220116,263568,313956,372096,438957,515508,602889,702240,814891,942172,1085623

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

%C Fourth column (m=3) of triangle A060098.

%C Partial sums of A038163.

%C 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]

%D 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.

%H Peter J. C. Moses, <a href="/A060099/b060099.txt">Table of n, a(n) for n = 0..9999</a>

%H Jia Huang, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Huang/huang8.html">Partially Palindromic Compositions</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 20.

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3,-8,14,0,-14,8,3,-4,1).

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

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

%t 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 *)

%t CoefficientList[Series[1/((1-x^2)^3*(1-x)^4),{x,0,100}],x] (* _Peter J. C. Moses_, Mar 24 2013 *)

%t 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 *)

%Y Cf. A002620, A002624, A096338.

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

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

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Apr 06 2001