cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A118401 Triangle, read by rows, equal to the matrix square of triangle A118400; also equals the matrix inverse of triangle A118407.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, -2, 2, 0, 1, 4, -2, 2, 0, 1, -6, 4, -2, 2, 0, 1, 8, -6, 4, -2, 2, 0, 1, -10, 8, -6, 4, -2, 2, 0, 1, 12, -10, 8, -6, 4, -2, 2, 0, 1, -14, 12, -10, 8, -6, 4, -2, 2, 0, 1, 16, -14, 12, -10, 8, -6, 4, -2, 2, 0, 1, -18, 16, -14, 12, -10, 8, -6, 4, -2, 2, 0, 1, 20, -18, 16, -14, 12, -10, 8, -6, 4, -2, 2, 0, 1
Offset: 0

Views

Author

Paul D. Hanna, Apr 27 2006

Keywords

Comments

This triangle has an integer matrix square-root (A118400) if the main diagonal of the square-root is allowed to be signed. Even though the columns of this triangle are all the same, the columns of the matrix square-root A118400 are all different.

Examples

			Triangle begins:
1;
0, 1;
2, 0, 1;
-2, 2, 0, 1;
4,-2, 2, 0, 1;
-6, 4,-2, 2, 0, 1;
8,-6, 4,-2, 2, 0, 1;
-10, 8,-6, 4,-2, 2, 0, 1;
12,-10, 8,-6, 4,-2, 2, 0, 1;
-14, 12,-10, 8,-6, 4,-2, 2, 0, 1;
16,-14, 12,-10, 8,-6, 4,-2, 2, 0, 1; ...

Crossrefs

Cf. A118400 (matrix square-root), A118402 (row sums), A118403 (unsigned row sums), A118407 (matrix inverse).

Programs

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

Formula

G.f.: A(x,y) = (1 + 2*x + 2*x^2)*(1+x^2)/(1+x)^2/(1-x*y). Column g.f.: (1 + 2*x + 2*x^2)*(1+x^2)/(1+x)^2.

A118403 Unsigned row sums of triangle A118401; a(n) = A118402(n^2-n+2), where A118402 is the row sums of triangle A118400.

Original entry on oeis.org

1, 1, 3, 5, 9, 15, 23, 33, 45, 59, 75, 93, 113, 135, 159, 185, 213, 243, 275, 309, 345, 383, 423, 465, 509, 555, 603, 653, 705, 759, 815, 873, 933, 995, 1059, 1125, 1193, 1263, 1335, 1409, 1485, 1563, 1643, 1725, 1809, 1895, 1983, 2073, 2165, 2259, 2355
Offset: 0

Views

Author

Paul D. Hanna, Apr 27 2006

Keywords

Links

Crossrefs

Cf. A027688, A118400, A118401, A118402.

Programs

  • Maple
    seq(coeff(series((1-2*x+2*x^2)*(1+x^2)/(1-x)^3,x,n+1), x, n), n = 0 .. 55); # Muniru A Asiru, Jan 02 2019
  • Mathematica
    Join[{1, 1}, Table[n^2 + n + 3, {n, 0, 47}]] (* Jon Maiga, Jan 02 2019 *)
  • PARI
    {a(n)=polcoeff((1-2*x+2*x^2)*(1+x^2)/(1-x+x*O(x^n))^3,n,x)}

Formula

G.f.: A(x) = (1-2*x+2*x^2)*(1+x^2)/(1-x)^3.
a(n) = A027688(n-2) for n > 1. - Jon Maiga, Jan 02 2019
E.g.f.: exp(x)*(5 - 2*x + x^2) - 2*(2 + x). - Stefano Spezia, Dec 21 2024

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.

Original entry on oeis.org

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

Views

Author

Paul D. Hanna, Apr 27 2006

Keywords

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

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

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.

Programs

  • 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.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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