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A118404 Triangle T, read by rows, where all columns of T are different and yet all columns of the matrix square T^2 (A118407) are equal; also equals the matrix inverse of triangle A118400. 4

%I #21 Jul 26 2018 14:28:34

%S 1,1,-1,-1,0,1,-1,1,-1,-1,1,0,0,2,1,1,-1,0,-2,-3,-1,-1,0,1,2,5,4,1,-1,

%T 1,-1,-3,-7,-9,-5,-1,1,0,0,4,10,16,14,6,1,1,-1,0,-4,-14,-26,-30,-20,

%U -7,-1,-1,0,1,4,18,40,56,50,27,8,1,-1,1,-1,-5,-22,-58,-96,-106,-77,-35,-9,-1,1,0,0,6,27,80,154,202,183,112,44,10,1,1,-1,0,-6,-33,-107,-234,-356,-385,-295,-156,-54,-11,-1,-1,0,1,6,39,140,341,590,741,680,451,210,65,12,1,-1,1,-1,-7,-45,-179,-481,-931,-1331,-1421,-1131,-661,-275,-77,-13,-1,1,0,0,8,52,224,660,1412,2262,2752,2552,1792,936,352,90,14,1

%N Triangle T, read by rows, where all columns of T are different and yet all columns of the matrix square T^2 (A118407) are equal; also equals the matrix inverse of triangle A118400.

%C Appears to coincide with triangle (5.2) in Lee-Oh (2016), although there is no obvious connection! - _N. J. A. Sloane_, Dec 07 2016

%H Kyu-Hwan Lee, Se-jin Oh, <a href="http://arxiv.org/abs/1601.06685">Catalan triangle numbers and binomial coefficients</a>, arXiv:1601.06685 [math.CO], 2016.

%F G.f.: A(x,y) = (1+x)^2 / ( (1+x^2) * (1+x + x*y) ).

%F G.f. of column k: (-1)^k / ( (1+x^2) * (1+x)^(k-1) ) for k>=0.

%e Triangle begins:

%e 1;

%e 1,-1;

%e -1, 0, 1;

%e -1, 1,-1,-1;

%e 1, 0, 0, 2, 1;

%e 1,-1, 0,-2,-3,-1;

%e -1, 0, 1, 2, 5, 4, 1;

%e -1, 1,-1,-3,-7,-9,-5,-1;

%e 1, 0, 0, 4, 10, 16, 14, 6, 1;

%e 1,-1, 0,-4,-14,-26,-30,-20,-7,-1;

%e -1, 0, 1, 4, 18, 40, 56, 50, 27, 8, 1;

%e -1, 1,-1,-5,-22,-58,-96,-106,-77,-35,-9,-1;

%e 1, 0, 0, 6, 27, 80, 154, 202, 183, 112, 44, 10, 1;

%e 1, -1, 0, -6, -33, -107, -234, -356, -385, -295, -156, -54, -11, -1;

%e -1, 0, 1, 6, 39, 140, 341, 590, 741, 680, 451, 210, 65, 12, 1;

%e -1, 1, -1, -7, -45, -179, -481, -931, -1331, -1421, -1131, -661, -275, -77, -13, -1;

%e 1, 0, 0, 8, 52, 224, 660, 1412, 2262, 2752, 2552, 1792, 936, 352, 90, 14, 1;

%e 1, -1, 0, -8, -60, -276, -884, -2072, -3674, -5014, -5304, -4344, -2728, -1288, -442, -104, -15, -1;

%e -1, 0, 1, 8, 68, 336, 1160, 2956, 5746, 8688, 10318, 9648, 7072, 4016, 1730, 546, 119, 16, 1; ...

%e The matrix square is A118407:

%e 1;

%e 0, 1;

%e -2, 0, 1;

%e 2,-2, 0, 1;

%e 0, 2,-2, 0, 1;

%e -2, 0, 2,-2, 0, 1;

%e 4,-2, 0, 2,-2, 0, 1;

%e -6, 4,-2, 0, 2,-2, 0, 1;

%e 4,-6, 4,-2, 0, 2,-2, 0, 1;

%e 6, 4,-6, 4,-2, 0, 2,-2, 0, 1; ...

%e in which all columns are equal.

%t T[n_, k_] := SeriesCoefficient[(-1)^k/((1+x^2)(1+x)^(k-1)), {x, 0, n-k}];

%t Table[T[n, k], {n, 0, 16}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 26 2018 *)

%o (PARI) {T(n,k)=polcoeff(polcoeff((1+x)^2/(1+x^2)/(1+x+x*y +x*O(x^n)),n,x)+y*O(y^k),k,y)}

%o for(n=0, 16, for(k=0, n, print1(T(n, k), ", ")); print(""))

%Y Cf. A118405 (row sums), A118406 (unsigned row sums), A118407 (matrix square), A118400 (matrix inverse).

%Y Columns or diagonals (modulo offsets): A219977, A011848, A212342, A007598, A005581, A007910.

%K sign,tabl

%O 0,14

%A _Paul D. Hanna_, Apr 27 2006

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