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%I #24 Aug 03 2022 07:33:51
%S 0,1,15,90,336,951,2247,4676,8856,15597,25927,41118,62712,92547,
%T 132783,185928,254864,342873,453663,591394,760704,966735,1215159,
%U 1512204,1864680,2280005,2766231,3332070,3986920,4740891,5604831,6590352
%N Rectified hexateron (5-simplex) numbers: the coefficient of x^(2n-2) in (1+x+x^2+...+x^(n-1))^6.
%C a(n) is the number of ordered 6-tuples (j_1,...,j_6) with 0 <= j_i <= n-1 and Sum_{i=1..6} j_i = 2n-2. - _Robert Israel_, Feb 17 2016
%H J. Conrad and Robert Israel, <a href="/A179096/b179096.txt">Table of n, a(n) for n = 0..1000</a> (n = 0..98 from J. Conrad)
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F Conjectures: a(n) = n*(n+1)*(13*n^3+12*n^2-7*n+12)/60. G.f.: x*(1+9*x+x^3+15*x^2)/(x-1)^6. - _R. J. Mathar_, Jul 06 2010
%F These conjectures are true, see A179095 for proof.
%p F:= n -> coeff(add(x^i,i=0..n-1)^6,x,2*n-2):
%p seq(F(n),n=0..100); # _Robert Israel_, Feb 17 2016
%t f[n_] := CoefficientList[ Series[ Sum[x^k, {k, 0, n - 1}]^6, {x, 0, 2 n + 3}], x][[2 n - 1]]; Array[f, 36] (* _Robert G. Wilson v_, Jul 30 2010 *)
%o (PARI) a(n) = polcoeff(((x^n-1)/(x-1))^6, 2*n-2); \\ _Michel Marcus_, Feb 17 2016
%Y Cf. A179095, A179097, A179098, A179099.
%K nonn,easy
%O 0,3
%A _Michael A. Jackson_, Jun 29 2010
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