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.
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, 1, -1, -3, -7, -9, -5, -1, 1, 0, 0, 4, 10, 16, 14, 6, 1, 1, -1, 0, -4, -14, -26, -30, -20, -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
Offset: 0
Examples
Triangle begins: 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, 1,-1,-3,-7,-9,-5,-1; 1, 0, 0, 4, 10, 16, 14, 6, 1; 1,-1, 0,-4,-14,-26,-30,-20,-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; 1, -1, 0, -8, -60, -276, -884, -2072, -3674, -5014, -5304, -4344, -2728, -1288, -442, -104, -15, -1; -1, 0, 1, 8, 68, 336, 1160, 2956, 5746, 8688, 10318, 9648, 7072, 4016, 1730, 546, 119, 16, 1; ... The matrix square is A118407: 1; 0, 1; -2, 0, 1; 2,-2, 0, 1; 0, 2,-2, 0, 1; -2, 0, 2,-2, 0, 1; 4,-2, 0, 2,-2, 0, 1; -6, 4,-2, 0, 2,-2, 0, 1; 4,-6, 4,-2, 0, 2,-2, 0, 1; 6, 4,-6, 4,-2, 0, 2,-2, 0, 1; ... in which all columns are equal.
Links
- Kyu-Hwan Lee, Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.
Crossrefs
Programs
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Mathematica
T[n_, k_] := SeriesCoefficient[(-1)^k/((1+x^2)(1+x)^(k-1)), {x, 0, n-k}]; Table[T[n, k], {n, 0, 16}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 26 2018 *) -
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)} for(n=0, 16, for(k=0, n, print1(T(n, k), ", ")); print(""))
Formula
G.f.: A(x,y) = (1+x)^2 / ( (1+x^2) * (1+x + x*y) ).
G.f. of column k: (-1)^k / ( (1+x^2) * (1+x)^(k-1) ) for k>=0.
Comments