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A103881 Square array T(n,k) (n >= 1, k >= 0) read by antidiagonals: coordination sequence for root lattice A_n. 26
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 (list; table; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..78.

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, Wojciech Szpankowski, On the Discrete Geometry of Differential Privacy via Ehrhart Theory, November 2017.

Arun Padakandla, P.R. Kumar, 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.

EXAMPLE

Array begins:

  1,  2,   2,    2,     2,     2,      2,      2, ...

  1,  6,  12,   18,    24,    30,     36,     42, ...

  1, 12,  42,   92,   162,   252,    362,    492, ...

  1, 20, 110,  340,   780,  1500,   2570,   4060, ...

  1, 30, 240, 1010,  2970,  7002,  14240,  26070, ...

  1, 42, 462, 2562,  9492, 27174,  65226, 137886, ...

  1, 56, 812, 5768, 26474, 91112, 256508, 623576, ...

  ...

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, 1, 12}, {k, 0, n-1}]] (* Jean-Fran├žois Alcover, Dec 27 2012 *)

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.

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

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Last modified July 21 10:07 EDT 2018. Contains 312850 sequences. (Running on oeis4.)