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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
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,14

COMMENTS

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

LINKS

Table of n, a(n) for n=0..152.

Kyu-Hwan Lee, Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.

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.

EXAMPLE

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.

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-Fran├žois Alcover, Jul 26 2018 *)

PROG

(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(""))

CROSSREFS

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

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

Sequence in context: A117479 A200650 A281743 * A089339 A249303 A319081

Adjacent sequences:  A118401 A118402 A118403 * A118405 A118406 A118407

KEYWORD

sign,tabl

AUTHOR

Paul D. Hanna, Apr 27 2006

STATUS

approved

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Last modified December 7 20:18 EST 2019. Contains 329847 sequences. (Running on oeis4.)