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
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