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a(n) = n*(4*n^6 + 70*n^4 + 196*n^2 + 45)/315.
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%I #40 Sep 17 2023 09:43:34

%S 0,1,14,99,476,1765,5418,14407,34232,74313,149830,284075,511380,

%T 880685,1459810,2340495,3644272,5529233,8197758,11905267,16970060,

%U 23784309,32826266,44673751,60018984,79684825,104642486,136030779,175176964,223619261,283131090,355747103

%N a(n) = n*(4*n^6 + 70*n^4 + 196*n^2 + 45)/315.

%C Kim asserts that every nonnegative integer can be represented by the sum of no more than 21 of these numbers.

%C Starting with 1 = binomial transform of [1, 13, 72, 220, 400, 432, 256, 0, 0, 0, ...], where (1, 13, 72, 220, 400, 432, 256) = row 7 of the Chebyshev triangle A081277. Also = row 7 of the array in A142978. - _Gary W. Adamson_, Jul 19 2008

%H Seiichi Manyama, <a href="/A099193/b099193.txt">Table of n, a(n) for n = 0..10000</a>

%H Milan Janjić, <a href="https://arxiv.org/abs/1905.04465">On Restricted Ternary Words and Insets</a>, arXiv:1905.04465 [math.CO], 2019.

%H Hyun Kwang Kim, <a href="http://dx.doi.org/10.1090/S0002-9939-02-06710-2">On Regular Polytope Numbers</a>, Proc. Amer. Math. Soc., 131 (2003), 65-75.

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8, -28, 56, -70, 56, -28, 8, -1).

%F a(n) = n*(4*n^6 + 70*n^4 + 196*n^2 + 45)/315.

%F G.f.: x*(1+x)^6/(1-x)^8. - _R. J. Mathar_, Jul 18 2009

%F a(n) = 14*a(n-1)/(n-1) + a(n-2) for n > 1. - _Seiichi Manyama_, Jun 06 2018

%t Table[SeriesCoefficient[x (1 + x)^6/(1 - x)^8, {x, 0, n}], {n, 0, 31}] (* _Michael De Vlieger_, Dec 14 2015 *)

%o (PARI) concat(0, Vec(x*(1+x)^6/(1-x)^8 + O(x^40))) \\ _Michel Marcus_, Dec 14 2015

%Y Similar sequences: A005900 (m=3), A014820(n-1) (m=4), A069038 (m=5), A069039 (m=6), A099195 (m=8), A099196 (m=9), A099197 (m=10).

%Y Cf. A000332.

%Y Cf. A142978, A081277.

%K easy,nonn

%O 0,3

%A _Jonathan Vos Post_, Nov 16 2004

%E More terms from _Michel Marcus_, Dec 14 2015