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Expansion of g.f. x*(1+x)^5/(1-x)^7.
14

%I #63 Mar 10 2024 14:55:17

%S 0,1,12,73,304,985,2668,6321,13504,26577,48940,85305,142000,227305,

%T 351820,528865,774912,1110049,1558476,2149033,2915760,3898489,5143468,

%U 6704017,8641216,11024625,13933036,17455257,21690928,26751369,32760460,39855553,48188416,57926209

%N Expansion of g.f. x*(1+x)^5/(1-x)^7.

%C Figurate numbers based on the 6-dimensional regular convex polytope called the 6-dimensional cross-polytope, or 6-dimensional hyperoctahedron, which is represented by the Schlaefli symbol {3, 3, 3, 3, 4}. It is the dual of the 6-dimensional hypercube. Kim asserts that every nonnegative integer can be represented by the sum of no more than 19 of these 6-crosspolytope numbers. - _Jonathan Vos Post_, Nov 16 2004

%C Starting with 1 = binomial transform of [1, 11, 50, 120, 160, 112, 32, 0, 0, 0, ...] where (1, 11, 50, 120, 160, 112, 32) = row 6 of the Chebyshev triangle A081277. Also = row 6 of the array in A142978. - _Gary W. Adamson_, Jul 19 2008

%D H. S. M. Coxeter, Regular Polytopes, New York: Dover, 1973.

%D E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 240.

%D Jonathan Vos Post, "4-Dimensional Jonathan numbers: polytope numbers and Centered polytope numbers of Higher Than 3 Dimensions", Draft 1.5 of 9 a.m., 12 March 2004, circulated by e-mail.

%H Seiichi Manyama, <a href="/A069039/b069039.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 Milan Janjic and B. Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv 1301.4550 [math.CO], 2013.

%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 Jonathan Vos Post, <a href="https://web.archive.org/web/20200219170305/http://magicdragon.com/poly.html">Table of polytope numbers, Sorted, Through 1,000,000</a>.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).

%F Recurrence: a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).

%F a(n) = (n^2)*(2*n^4 + 20*n^2 + 23 )/45. - _Jonathan Vos Post_, Nov 16 2004

%F From _Stephen Crowley_, Jul 14 2009: (Start)

%F Sum_{n >= 1} 1/a(n) = -5*(Sum(_alpha*(77*_alpha^2+655)*Psi(1-_alpha), _alpha = RootOf(2*_Z^4+20*_Z^2+23)))*(1/3174)+15*Pi^2*(1/46)=1.10203455013915915542552577192042916250524...

%F Sum_{n>=1} 1/(a(n)*n!) = hypergeom([1, 1, 1, 1-a, 1+b, 1-b, 1+a], [2, 2, 2, 2+b, 2-b, 2+a, 2-a], 1) = 1.04409584723862654376639417281585634150689... where a = (i/2)*sqrt(20+6*sqrt(6)), b = (i/2)*sqrt(20-6*sqrt(6)), and i = sqrt(-1). (End)

%F a(n) = 12*a(n-1)/(n-1) + a(n-2) for n > 1. - _Seiichi Manyama_, Jun 06 2018

%F E.g.f.: exp(x)*x*(45 + 225*x + 300*x^2 + 150*x^3 + 30*x^4 + 2*x^5)/45. - _Stefano Spezia_, Mar 10 2024

%p al:=proc(s,n) binomial(n+s-1,s); end; be:=proc(d,n) local r; add( (-1)^r*binomial(d-1,r)*2^(d-1-r)*al(d-r,n), r=0..d-1); end; [seq(be(6,n),n=0..100)];

%t a[n_] := n^2*(2*n^4 + 20*n^2 + 23)/45; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jan 29 2014 *)

%t CoefficientList[Series[x (1+x)^5/(1-x)^7,{x,0,40}],x] (* or *) LinearRecurrence[ {7,-21,35,-35,21,-7,1},{0,1,12,73,304,985,2668},40] (* _Harvey P. Dale_, Aug 05 2018 *)

%o (PARI) x='x+O('x^100); concat(0, Vec(x*(1+x)^5/(1-x)^7)) \\ _Altug Alkan_, Dec 14 2015

%Y Similar sequence: A005900 (m=3), A014820(n-1) (m=4), A069038 (m=5), A099193 (m=7), A099195 (m=8), A099196 (m=9), A099197 (m=10).

%Y Cf. A000332.

%Y Cf. A081277, A142978.

%K nonn,easy

%O 0,3

%A _Vladeta Jovovic_, Apr 03 2002