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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]

REFERENCES

J. Serra-Sagrista, Enumeration of lattice points in l_1 norm, Information Processing Letters, 76, no. 1-2 (2000), 39-44.

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).

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 September 24 18:54 EDT 2017. Contains 292433 sequences.