A103884 - OEIS

%I #31 Jul 09 2023 12:14:48

%S 1,1,8,1,18,16,1,32,66,24,1,50,192,146,32,1,72,450,608,258,40,1,98,

%T 912,1970,1408,402,48,1,128,1666,5336,5890,2720,578,56,1,162,2816,

%U 12642,20256,14002,4672,786,64,1,200,4482,27008,59906,58728,28610,7392,1026,72

%N Square array A(n,k) read by antidiagonals: row n gives coordination sequence for lattice C_n.

%H G. C. Greubel, <a href="/A103884/b103884.txt">Antidiagonals n = 2..50, flattened</a>

%H M. Baake and U. Grimm, <a href="http://arXiv.org/abs/cond-mat/9706122">Coordination sequences for root lattices and related graphs</a>, arXiv:cond-mat/9706122 [cond-mat.stat-mech], 1997.

%H J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (<a href="http://neilsloane.com/doc/Me220.pdf">pdf</a>).

%H Joan Serra-Sagrista, <a href="https://dx.doi.org/10.1016/S0020-0190(00)00119-8">Enumeration of lattice points in l_1 norm</a>, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.

%F A(n,k) = Sum_{i=1..2*k} 2^i*C(n, i)*C(2*k-1, i-1), A(n,0) = 1 (array).

%F G.f. of n-th row: (Sum_{i=0..n} C(2*n, 2*i)*x^i)/(1-x)^n.

%F T(n, k) = A(n-k, k) (antidiagonals).

%F T(n, n-2) = A022144(n-2).

%F 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

%F From  _Peter Bala_, Jul 09 2023: (Start)

%F 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.

%F 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)

%e Array begins:

%e   1,   8,    16,     24,      32,       40,        48, ... A022144;

%e   1,  18,    66,    146,     258,      402,       578, ... A010006;

%e   1,  32,   192,    608,    1408,     2720,      4672, ... A019560;

%e   1,  50,   450,   1970,    5890,    14002,     28610, ... A019561;

%e   1,  72,   912,   5336,   20256,    58728,    142000, ... A019562;

%e   1,  98,  1666,  12642,   59906,   209762,    596610, ... A019563;

%e   1, 128,  2816,  27008,  157184,   658048,   2187520, ... A019564;

%e   1, 162,  4482,  53154,  374274,  1854882,   7159170, ... A035746;

%e   1, 200,  6800,  97880,  822560,  4780008,  21278640, ... A035747;

%e   1, 242,  9922, 170610, 1690370, 11414898,  58227906, ... A035748;

%e   1, 288, 14016, 284000, 3281280, 25534368, 148321344, ... A035749;

%e   ...

%e Antidiagonals, T(n, k), begins as:

%e   1;

%e   1,   8;

%e   1,  18,   16;

%e   1,  32,   66,   24;

%e   1,  50,  192,  146,   32;

%e   1,  72,  450,  608,  258,   40;

%e   1,  98,  912, 1970, 1408,  402,  48;

%e   1, 128, 1666, 5336, 5890, 2720, 578, 56;

%t 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 *)

%o (Magma)

%o 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]]) >;

%o [A103884(n,k): k in [0..n-2], n in [2..12]]; // _G. C. Greubel_, May 23 2023

%o (SageMath)

%o 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))

%o flatten([[A103884(n,k) for k in range(n-1)] for n in range(2,13)]) # _G. C. Greubel_, May 23 2023

%Y Rows include A022144, A010006, A019560, A019561, A019562, A019563, A019564, A035746, A035747, A035748, A035749, A035750 - A035787.

%Y Columns include A001105, A035598, A035600, A035602, A035604, A035606.

%Y Main diagonal is in A103885.

%Y Cf. A103881, A103993, A108998.

%K nonn,tabl

%O 2,3

%A _Ralf Stephan_, Feb 20 2005

%E Definition clarified by _N. J. A. Sloane_, May 25 2023