A008395 -id:A008395 - cp's OEIS frontend

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

Ralf Stephan, Feb 20 2005

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

			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;

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.

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.
Cf. A047085, A103884, A103903, A103998, A143007.

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

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

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:

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 *)

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

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

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)

Corrected by N. J. A. Sloane, Dec 15 2012, at the suggestion of Manuel Blum

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A035836 Duplicate of A008395.

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A103881 Square array T(n,k) (n >= 1, k >= 0) read by antidiagonals: coordination sequence for root lattice A_n.

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