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
Examples
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;
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.
Crossrefs
Programs
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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
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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
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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:
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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 *) -
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
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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
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)
Extensions
Corrected by N. J. A. Sloane, Dec 15 2012, at the suggestion of Manuel Blum
Comments