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%I #69 Jul 09 2023 12:14:43
%S 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,
%T 30,2,1,56,462,1010,780,252,36,2,1,72,812,2562,2970,1500,362,42,2,1,
%U 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
%N Square array T(n,k) (n >= 1, k >= 0) read by antidiagonals: coordination sequence for root lattice A_n.
%C 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
%H Muniru A Asiru, <a href="/A103881/b103881.txt">Table of n, a(n) for n = 1..5050</a> (antidiagonals 1 to 100, flattened)
%H M. Baake and U. Grimm, <a href="http://arXiv.org/abs/cond-mat/9706122">Coordination sequences for root lattices and related graphs</a>, arXiv:cond-mat/9706122 [cond-mat.stat-mech], 1997.
%H J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (<a href="http://neilsloane.com/doc/Me220.pdf">pdf</a>).
%H Arun Padakandla, P. R. Kumar, and Wojciech Szpankowski, <a href="https://www.cs.purdue.edu/homes/arunpr/preprints/DiscreteGeometryViaEhrhart.pdf">On the Discrete Geometry of Differential Privacy via Ehrhart Theory</a>, November 2017.
%H Arun Padakandla, P. R. Kumar, and Wojciech Szpankowski, <a href="https://www.cs.purdue.edu/homes/spa/papers/soda-privacy.pdf">Preserving Privacy and Fidelity via Ehrhart Theory</a>, July 2017.
%H Joan Serra-Sagrista, <a href="http://dx.doi.org/10.1016/S0020-0190(00)00119-8">Enumeration of lattice points in l_1 norm</a>, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.
%F 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.
%F G.f. of n-th row: (Sum_{i=0..n} C(n, i)^2*x^i)/(1-x)^n.
%F From _G. C. Greubel_, May 24 2023: (Start)
%F T(n, k) = Sum_{j=0..n} binomial(n,j)^2 * binomial(n+k-j-1, n-1) (array).
%F 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).
%F 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).
%F Sum_{k=0..n-1} t(n, k) = A047085(n). (End)
%F From _Peter Bala_, Jul 09 2023: (Start)
%F T(n,k) = [x^k] Legendre_P(n, (1 + x)/(1 - x)).
%F (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)
%e Array begins:
%e 1, 2, 2, 2, 2, 2, 2, 2, ... A040000;
%e 1, 6, 12, 18, 24, 30, 36, 42, ... A008458;
%e 1, 12, 42, 92, 162, 252, 362, 492, ... A005901;
%e 1, 20, 110, 340, 780, 1500, 2570, 4060, ... A008383;
%e 1, 30, 240, 1010, 2970, 7002, 14240, 26070, ... A008385;
%e 1, 42, 462, 2562, 9492, 27174, 65226, 137886, ... A008387;
%e 1, 56, 812, 5768, 26474, 91112, 256508, 623576, ... A008389;
%e 1, 72, 1332, 11832, 66222, 271224, 889716, 2476296, ... A008391;
%e 1, 90, 2070, 22530, 151560, 731502, 2777370, 8809110, ... A008393;
%e 1, 110, 3080, 40370, 322190, 1815506, 7925720, 28512110, ... A008395;
%e 1, 132, 4422, 68772, 643632, 4197468, 20934474, 85014204, ... A035837;
%e 1, 156, 6162, 112268, 1219374, 9129276, 51697802, 235895244, ... A035838;
%e 1, 182, 8372, 176722, 2206932, 18827718, 120353324, 614266354, ... A035839;
%e 1, 210, 11130, 269570, 3838590, 37060506, 265953170, 1511679210, ... A035840;
%e ...
%e Antidiagonals:
%e 1;
%e 1, 2;
%e 1, 6, 2;
%e 1, 12, 12, 2;
%e 1, 20, 42, 18, 2;
%e 1, 30, 110, 92, 24, 2;
%e 1, 42, 240, 340, 162, 30, 2;
%e 1, 56, 462, 1010, 780, 252, 36, 2;
%e 1, 72, 812, 2562, 2970, 1500, 362, 42, 2;
%e 1, 90, 1332, 5768, 9492, 7002, 2570, 492, 48, 2;
%p T:=proc(n,k) option remember; local i;
%p if k=0 then 1 else
%p add( binomial(n+1,i)*binomial(k-1,i-1)*binomial(n-i+k,k),i=1..n); fi;
%p end:
%p g:=n->[seq(T(n-i,i),i=0..n-1)]:
%p for n from 1 to 14 do lprint(op(g(n))); od:
%t 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 *)
%o (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
%o (PARI)
%o 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)));
%o for(n=1, 15, for(k=0, n-1, print1(A103881(n,k), ", "))) \\ _G. C. Greubel_, Oct 16 2018; May 24 2023
%o (Magma)
%o 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]]) >;
%o [A103881(n,k): k in [0..n-1], n in [1..15]]; // _G. C. Greubel_, Oct 16 2018; May 24 2023
%o (SageMath)
%o 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()
%o flatten([[A103881(n,k) for k in range(n)] for n in range(1,16)]) # _G. C. Greubel_, May 24 2023
%Y Rows include A040000, A008458, A005901, A008383, A008385, A008387, A008389, A008391, A008393, A008395, A035837, A035838, A035839, A035840, A035841 - A035876.
%Y Columns include A002376, A001621.
%Y Main diagonal is in A103882.
%Y Cf. A047085, A103884, A103903, A103998, A143007.
%K nonn,tabl
%O 1,3
%A _Ralf Stephan_, Feb 20 2005
%E Corrected by _N. J. A. Sloane_, Dec 15 2012, at the suggestion of Manuel Blum
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