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 1-4 of 4 results.

A118402 Row sums of triangle A118401.

Original entry on oeis.org

1, 1, 3, 1, 5, -1, 7, -3, 9, -5, 11, -7, 13, -9, 15, -11, 17, -13, 19, -15, 21, -17, 23, -19, 25, -21, 27, -23, 29, -25, 31, -27, 33, -29, 35, -31, 37, -33, 39, -35, 41, -37, 43, -39, 45, -41, 47, -43, 49, -45, 51, -47, 53, -49, 55, -51, 57, -53, 59, -55, 61, -57, 63, -59, 65, -61, 67, -63, 69, -65, 71
Offset: 0

Views

Author

Paul D. Hanna, Apr 27 2006

Keywords

Links

Crossrefs

Cf. A118401, A118403.

Programs

  • Maple
    a := n -> `if`(n=1, 1, (5+(-1)^n*(2*n-3))/2);
seq(a(n), n=0..70); # Peter Luschny, Aug 04 2014
  • Mathematica
    Join[{1, 1}, LinearRecurrence[{-1, 1, 1}, {3, 1, 5}, 70]] (* Jean-François Alcover, Jun 13 2019 *)
  • PARI
    {a(n)=polcoeff((1+2*x+2*x^2)*(1+x^2)/(1+x+x*O(x^n))^2/(1-x),n,x)}

Formula

G.f.: (1+2*x+2*x^2)*(1+x^2)/(1+x)^2/(1-x).
a(n^2-n+2) = A118403(n).
a(2n) = 2n+1, a(2n+1) = 3-2n, n>0. - Ralf Stephan, Aug 18 2013
a(n) = (5+(-1)^n*(2*n-3))/2 for n>1. - Peter Luschny, Aug 04 2014

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

A118407 Triangle, read by rows, equal to the matrix square of triangle A118404; also equals the matrix inverse of triangle A118401.

Original entry on oeis.org

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, -20, 6, 4, -6, 4, -2, 0, 2, -2, 0, 1, 26, -20, 6, 4, -6, 4, -2, 0, 2, -2, 0, 1, -12, 26, -20, 6, 4, -6, 4, -2, 0, 2, -2, 0, 1
Offset: 0

Views

Author

Paul D. Hanna, Apr 27 2006

Keywords

Comments

This triangle has an integer matrix square-root (A118404) 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 A118404 are all different.

Examples

			Triangle begins:
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;
-20, 6, 4,-6, 4,-2, 0, 2,-2, 0, 1;
26,-20, 6, 4,-6, 4,-2, 0, 2,-2, 0, 1; ...

Crossrefs

Cf. A118404 (matrix square-root), A118401 (matrix inverse), A118408 (row sums), A118409 (unsigned row sums).

Programs

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

Formula

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

A118400 Triangle T, read by rows, where all columns of T are different and yet all columns of the matrix square T^2 (A118401) are equal; a signed version of triangle A087698.

Original entry on oeis.org

1, 1, -1, 1, 0, 1, -1, -1, -1, -1, 1, 2, 2, 2, 1, -1, -3, -4, -4, -3, -1, 1, 4, 7, 8, 7, 4, 1, -1, -5, -11, -15, -15, -11, -5, -1, 1, 6, 16, 26, 30, 26, 16, 6, 1, -1, -7, -22, -42, -56, -56, -42, -22, -7, -1, 1, 8, 29, 64, 98, 112, 98, 64, 29, 8, 1, -1, -9, -37, -93, -162, -210, -210, -162, -93, -37, -9, -1
Offset: 0

Views

Author

Paul D. Hanna, Apr 27 2006

Keywords

Comments

Matrix inverse equals A118404. Row sums equal A084633. Signed version of: A087698 = maximum number of Boolean inputs at Hamming distance 2 for symmetric Boolean functions. This is an example of the fact that special matrices (cf. A118401) can have more than 2 signed matrix square-roots if the main diagonal is allowed to be signed.

Examples

			Triangle T begins:
1;
1,-1;
1, 0, 1;
-1,-1,-1,-1;
1, 2, 2, 2, 1;
-1,-3,-4,-4,-3,-1;
1, 4, 7, 8, 7, 4, 1;
-1,-5,-11,-15,-15,-11,-5,-1;
1, 6, 16, 26, 30, 26, 16, 6, 1;
-1,-7,-22,-42,-56,-56,-42,-22,-7,-1;
1, 8, 29, 64, 98, 112, 98, 64, 29, 8, 1;
-1,-9,-37,-93,-162,-210,-210,-162,-93,-37,-9,-1; ...
The matrix square is A118401:
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; ...
in which all columns are equal.

Crossrefs

Cf. A118401 (matrix square), A084633 (row sums), A087698 (unsigned version); A118404 (matrix inverse).

Programs

  • PARI
    T(n,k)=polcoeff(polcoeff((1+2*x+2*x^2)/(1+x+x*y+x*O(x^n)),n,x)+y*O(y^k),k,y)
  • PARI
    T(n,k)=if(n==1&k==0,1,(-1)^n*(binomial(n,k)-2*binomial(n-2,k-1)))

Formula

G.f.: A(x,y) = (1+2*x+2*x^2)/(1+x+x*y). G.f. of column k = (-1)^k*(1+2*x+2*x^2)/(1+x)^(k+1) for k>=0. T(n,k) = (-1)^n*[C(n,k) - 2*C(n-2,k-1)] for n>=k>=0 except that T(1,0)=1.
Showing 1-4 of 4 results.

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