A035879 Coordination sequence for diamond structure D^+_6. (Edges defined by l_1 norm = 1.)

1, 0, 72, 32, 1104, 672, 7128, 4032, 28320, 14784, 84072, 41184, 206064, 96096, 441336, 198016, 855360, 372096, 1535112, 651168, 2592144, 1076768, 4165656, 1700160, 6425568, 2583360, 9575592, 3800160, 13856304, 5437152, 19548216, 7594752, 26974848

Offset: 0

Author

Joan Serra-Sagrista (jserra(AT)ccd.uab.es)

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
• Joan Serra-Sagrista, Enumeration of lattice points in l_1 norm, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.
• Index entries for linear recurrences with constant coefficients, signature (0,6,0,-15,0,20,0,-15,0,6,0,-1).

Programs

Magma

[1] cat [2*n*((2*n^4+20*n^2+23)*(-1)^n+(4*n^4+10*n^2+31))/15: n in [1..20]]; // Bruno Berselli, Oct 21 2013

Maple

f := proc(m) local k,t1; t1 := 2^(n-1)*binomial((n+2*m)/2-1,n-1); if m mod 2 = 0 then t1 := t1+add(2^k*binomial(n,k)*binomial(m-1,k-1),k=0..n); fi; t1; end; where n=6.

Mathematica

f[m_, n_] := 2^(n - 1)*Binomial[(n + 2*m)/2 - 1, n - 1] + If[EvenQ[m], 2*n*Hypergeometric2F1[1 - m, 1 - n, 2, 2], 0]; f[0, ] = 1; Table[f[m, 6], {m, 0, 27}] (* _Jean-François Alcover, Apr 18 2013, after Maple *)
CoefficientList[Series[(x^12 + 66 x^10 + 32 x^9 + 687 x^8 + 480 x^7 + 1564 x^6 + 480 x^5 + 687 x^4 + 32 x^3 + 66 x^2 + 1)/((x - 1)^6 (x + 1)^6), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 21 2013 *)

Formula

G.f.: (x^12 + 66*x^10 + 32*x^9 + 687*x^8 + 480*x^7 + 1564*x^6 + 480*x^5 + 687*x^4 + 32*x^3 + 66*x^2 + 1) / ((x-1)^6*(x+1)^6). [Colin Barker, Nov 20 2012]

a(n) = 2*n*( (2*n^4+20*n^2+23)*(-1)^n + (4*n^4+10*n^2+31) )/15 for n>0, a(0)=1. [Bruno Berselli, Oct 21 2013]

Extensions

Recomputed by N. J. A. Sloane, Nov 27 1998

More terms from Vincenzo Librandi, Oct 21 2013