A103881 Square array T(n,k) (n >= 1, k >= 0) read by antidiagonals: coordination sequence for root lattice A_n.

1, 1, 2, 1, 6, 2, 1, 12, 12, 2, 1, 20, 42, 18, 2, 1, 30, 110, 92, 24, 2, 1, 42, 240, 340, 162, 30, 2, 1, 56, 462, 1010, 780, 252, 36, 2, 1, 72, 812, 2562, 2970, 1500, 362, 42, 2, 1, 90, 1332, 5768, 9492, 7002, 2570, 492, 48, 2, 1, 110, 2070, 11832, 26474, 27174, 14240, 4060, 642, 54, 2, 1, 132, 3080, 22530, 66222, 91112, 65226, 26070, 6040, 812, 60, 2

Offset

1,3

Comments

T(n,k) is the number of integer sequences of length n+1 with sum zero and sum of absolute values 2k. - R. H. Hardin, Feb 23 2009

Links

Muniru A Asiru, Table of n, a(n) for n = 1..5050 (antidiagonals 1 to 100, 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).

Arun Padakandla, P. R. Kumar, and Wojciech Szpankowski, On the Discrete Geometry of Differential Privacy via Ehrhart Theory, November 2017.

Arun Padakandla, P. R. Kumar, and Wojciech Szpankowski, Preserving Privacy and Fidelity via Ehrhart Theory, July 2017.

Joan Serra-Sagrista, Enumeration of lattice points in l_1 norm, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.

Formula

T(n,k) = Sum_{i=1..n} C(n+1, i)*C(k-1, i-1)*C(n-i+k, k), T(n,0)=1.

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

From G. C. Greubel, May 24 2023: (Start)

T(n, k) = Sum_{j=0..n} binomial(n,j)^2 * binomial(n+k-j-1, n-1) (array).

T(n, k) = (n+1)*binomial(n+k-1,k)*hypergeometric([-n,1-n,1-k], [2,1-n-k], 1), with T(n, k) = 1 (array).

t(n, k) = (n-k+1)*binomial(n-1,k)*hypergeometric([k-n,1+k-n,1-k], [2,1-n], 1), with t(n, 0) = 1 (antidiagonals).

Sum_{k=0..n-1} t(n, k) = A047085(n). (End)

From Peter Bala, Jul 09 2023: (Start)

T(n,k) = [x^k] Legendre_P(n, (1 + x)/(1 - x)).

(n+1)*T(n+1,k) = (n+1)*T(n+1,k-1) + (2*n+1)*(T(n,k) + T(n,k-1)) - n*(T(n-1,k) - T(n-1,k-1)). (End)

Example

Array begins:
  1,   2,     2,      2,       2,        2,         2,          2, ... A040000;
  1,   6,    12,     18,      24,       30,        36,         42, ... A008458;
  1,  12,    42,     92,     162,      252,       362,        492, ... A005901;
  1,  20,   110,    340,     780,     1500,      2570,       4060, ... A008383;
  1,  30,   240,   1010,    2970,     7002,     14240,      26070, ... A008385;
  1,  42,   462,   2562,    9492,    27174,     65226,     137886, ... A008387;
  1,  56,   812,   5768,   26474,    91112,    256508,     623576, ... A008389;
  1,  72,  1332,  11832,   66222,   271224,    889716,    2476296, ... A008391;
  1,  90,  2070,  22530,  151560,   731502,   2777370,    8809110, ... A008393;
  1, 110,  3080,  40370,  322190,  1815506,   7925720,   28512110, ... A008395;
  1, 132,  4422,  68772,  643632,  4197468,  20934474,   85014204, ... A035837;
  1, 156,  6162, 112268, 1219374,  9129276,  51697802,  235895244, ... A035838;
  1, 182,  8372, 176722, 2206932, 18827718, 120353324,  614266354, ... A035839;
  1, 210, 11130, 269570, 3838590, 37060506, 265953170, 1511679210, ... A035840;
  ...
Antidiagonals:
  1;
  1,  2;
  1,  6,    2;
  1, 12,   12,    2;
  1, 20,   42,   18,    2;
  1, 30,  110,   92,   24,    2;
  1, 42,  240,  340,  162,   30,    2;
  1, 56,  462, 1010,  780,  252,   36,   2;
  1, 72,  812, 2562, 2970, 1500,  362,  42,  2;
  1, 90, 1332, 5768, 9492, 7002, 2570, 492, 48,  2;

Maple

T:=proc(n, k) option remember; local i;
if k=0 then 1 else
add( binomial(n+1, i)*binomial(k-1, i-1)*binomial(n-i+k, k), i=1..n); fi;
end:
g:=n->[seq(T(n-i, i), i=0..n-1)]:
for n from 1 to 14 do lprint(op(g(n))); od:

Mathematica

T[n_, k_]:= (n+1)*(n+k-1)!*HypergeometricPFQ[{1-k, 1-n, -n}, {2, -n-k+1}, 1]/(k!*(n-1)!); T[_, 0]=1; Flatten[Table[T[n-k, k], {n, 12}, {k, 0, n-1}]] (* Jean-François Alcover, Dec 27 2012 *)

Prog

(GAP) T:=Flat(List([1..12], n->Concatenation([1], List([1..n-1], k->Sum([1..n], i->Binomial(n-k+1, i)*Binomial(k-1, i-1)*Binomial(n-i, k)))))); # Muniru A Asiru, Oct 14 2018
(PARI)
A103881(n, k) = if(k==0, 1, sum(j=1, n-k, binomial(n-k+1, j)*binomial(k-1, j-1)*binomial(n-j, k)));
for(n=1, 15, for(k=0, n-1, print1(A103881(n, k), ", "))) \\ G. C. Greubel, Oct 16 2018; May 24 2023
(Magma)
A103881:= func< n, k | k le 0 select 1 else (&+[Binomial(n-k+1, j)*Binomial(k-1, j-1)*Binomial(n-j, k): j in [1..n-k]]) >;
[A103881(n, k): k in [0..n-1], n in [1..15]]; // G. C. Greubel, Oct 16 2018; May 24 2023
(SageMath)
def A103881(n, k): return 1 if k==0 else (n-k+1)*binomial(n-1, k)*hypergeometric([k-n, 1+k-n, 1-k], [2, 1-n], 1).simplify()
flatten([[A103881(n, k) for k in range(n)] for n in range(1, 16)]) # G. C. Greubel, May 24 2023

Crossrefs

Rows include A040000, A008458, A005901, A008383, A008385, A008387, A008389, A008391, A008393, A008395, A035837, A035838, A035839, A035840, A035841 - A035876.

Columns include A002376, A001621.

Main diagonal is in A103882.

Cf. A047085, A103884, A103903, A103998, A143007.

Sequence in context: A208749 A208751 A133200 * A101024 A124730 A114283

Adjacent sequences: A103878 A103879 A103880 * A103882 A103883 A103884

Keyword

nonn,tabl

Author

Ralf Stephan, Feb 20 2005

Extensions

Corrected by N. J. A. Sloane, Dec 15 2012, at the suggestion of Manuel Blum

Status

approved