A103884 Square array A(n,k) read by antidiagonals: row n gives coordination sequence for lattice C_n.
1, 1, 8, 1, 18, 16, 1, 32, 66, 24, 1, 50, 192, 146, 32, 1, 72, 450, 608, 258, 40, 1, 98, 912, 1970, 1408, 402, 48, 1, 128, 1666, 5336, 5890, 2720, 578, 56, 1, 162, 2816, 12642, 20256, 14002, 4672, 786, 64, 1, 200, 4482, 27008, 59906, 58728, 28610, 7392, 1026, 72
Offset: 2
Examples
Array begins: 1, 8, 16, 24, 32, 40, 48, ... A022144; 1, 18, 66, 146, 258, 402, 578, ... A010006; 1, 32, 192, 608, 1408, 2720, 4672, ... A019560; 1, 50, 450, 1970, 5890, 14002, 28610, ... A019561; 1, 72, 912, 5336, 20256, 58728, 142000, ... A019562; 1, 98, 1666, 12642, 59906, 209762, 596610, ... A019563; 1, 128, 2816, 27008, 157184, 658048, 2187520, ... A019564; 1, 162, 4482, 53154, 374274, 1854882, 7159170, ... A035746; 1, 200, 6800, 97880, 822560, 4780008, 21278640, ... A035747; 1, 242, 9922, 170610, 1690370, 11414898, 58227906, ... A035748; 1, 288, 14016, 284000, 3281280, 25534368, 148321344, ... A035749; ... Antidiagonals, T(n, k), begins as: 1; 1, 8; 1, 18, 16; 1, 32, 66, 24; 1, 50, 192, 146, 32; 1, 72, 450, 608, 258, 40; 1, 98, 912, 1970, 1408, 402, 48; 1, 128, 1666, 5336, 5890, 2720, 578, 56;
Links
- G. C. Greubel, Antidiagonals n = 2..50, flattened
- M. Baake and U. Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122 [cond-mat.stat-mech], 1997.
- 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.
Crossrefs
Programs
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Magma
A103884:= func< n,k | k eq 0 select 1 else 2*(&+[2^j*Binomial(n-k,j+1)*Binomial(2*k-1,j) : j in [0..2*k-1]]) >; [A103884(n,k): k in [0..n-2], n in [2..12]]; // G. C. Greubel, May 23 2023
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Mathematica
nmin = 2; nmax = 11; t[n_, 0]= 1; t[n_, k_]:= 2n*Hypergeometric2F1[1-2k, 1-n, 2, 2]; tnk= Table[ t[n, k], {n, nmin, nmax}, {k, 0, nmax-nmin}]; Flatten[ Table[ tnk[[ n-k+1, k ]], {n, 1, nmax-nmin+1}, {k, 1, n} ] ] (* Jean-François Alcover, Jan 24 2012, after formula *) -
SageMath
def A103884(n,k): return 1 if k==0 else 2*sum(2^j*binomial(n-k,j+1)*binomial(2*k-1,j) for j in range(2*k)) flatten([[A103884(n,k) for k in range(n-1)] for n in range(2,13)]) # G. C. Greubel, May 23 2023
Formula
A(n,k) = Sum_{i=1..2*k} 2^i*C(n, i)*C(2*k-1, i-1), A(n,0) = 1 (array).
G.f. of n-th row: (Sum_{i=0..n} C(2*n, 2*i)*x^i)/(1-x)^n.
T(n, k) = A(n-k, k) (antidiagonals).
T(n, n-2) = A022144(n-2).
T(n, k) = 2*(n-k)*Hypergeometric2F1([1+k-n, 1-2*k], [2], 2), T(n, 0) = 1, for n >= 2, 0 <= k <= n-2. - G. C. Greubel, May 23 2023
From Peter Bala, Jul 09 2023: (Start)
T(n,k) = [x^k] Chebyshev_T(n, (1 + x)/(1 - x)), where Chebyshev_T(n, x) denotes the n-th Chebyshev polynomial of the first kind.
T(n+1,k) = T(n+1,k-1) + 2*T(n,k) + 2*T(n,k-1) + T(n-1,k) - T(n-1,k-1). (End)
Extensions
Definition clarified by N. J. A. Sloane, May 25 2023