A035881 Coordination sequence for diamond structure D^+_10. (Edges defined by l_1 norm = 1.)
1, 0, 200, 0, 6800, 512, 103000, 28160, 935200, 366080, 5805032, 2562560, 27135920, 12446720, 102442360, 47297536, 328075840, 150492160, 922953480, 418401280, 2339194064, 1046003200, 5442091160, 2399654400, 11788144480, 5127682560, 24036948520, 10321958400
Offset: 0
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 (0,10,0,-45,0,120,0,-210,0,252,0,-210,0,120,0,-45,0,10,0,-1).
Programs
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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=10.
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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, 10], {m, 0, 22}] (* _Jean-François Alcover, Apr 18 2013, after Maple *) CoefficientList[Series[(x^20 + 190 x^18 + 4845 x^16 + 512 x^15 + 43880 x^14 + 23040 x^13 + 187410 x^12 + 107520 x^11 + 313780 x^10 + 107520 x^9 + 187410 x^8 + 23040 x^7 + 43880 x^6 + 512 x^5 + 4845 x^4 + 190 x^2 + 1)/((x - 1)^10 (x + 1)^10), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 21 2013 *)
Formula
G.f.: (x^20 +190*x^18 +4845*x^16 +512*x^15 +43880*x^14 +23040*x^13 +187410*x^12 +107520*x^11 +313780*x^10 +107520*x^9 +187410*x^8 +23040*x^7 +43880*x^6 +512*x^5 +4845*x^4 +190*x^2 +1) / (( x -1)^10*( x +1)^10). [Colin Barker, Nov 20 2012]
Extensions
Recomputed by N. J. A. Sloane, Nov 27 1998
More terms from Vincenzo Librandi, Oct 21 2013