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.

Previous Showing 31-34 of 34 results.

A081281 Square array of binomial transforms of Chebyshev polynomial coefficients.

Original entry on oeis.org

1, 1, 2, 1, 4, 5, 1, 6, 15, 14, 1, 8, 29, 54, 41, 1, 10, 47, 126, 189, 122, 1, 12, 69, 238, 513, 648, 365, 1, 14, 95, 398, 1101, 1998, 2187, 1094, 1, 16, 125, 614, 2057, 4788, 7533, 7290, 3281, 1, 18, 159, 894, 3501, 9858, 19899, 27702, 24057, 9842, 1, 20, 197, 1246
Offset: 0

Views

Author

Paul Barry, Mar 16 2003

Keywords

Comments

Square array of binomial transforms of rows of A081277. Rows include A007051, A006234, A081278, A081279, A081280.

Examples

			Read by antidiagonals, the array appears as {1}, {1,2},{1,4,5}, {1,6,15,14}, {1,8,29,54,41},...

Formula

Rows have g.f. (1-2x)(1-x)^(n-1)/(1-3x)^(n+1).

A103407 Triangle of absolute values of the coefficients (in descending powers) of the characteristic polynomials of n X n matrices with 3's on the main diagonal and 1's elsewhere.

Original entry on oeis.org

1, 1, 3, 1, 6, 8, 1, 9, 24, 20, 1, 12, 48, 80, 48, 1, 15, 80, 200, 240, 112, 1, 18, 120, 400, 720, 672, 256, 1, 21, 168, 700, 1680, 2352, 1792, 576, 1, 24, 224, 1120, 3360, 6272, 7168, 4608, 1280, 1, 27, 288, 1680, 6048, 14112, 21504, 20736, 11520, 2816, 1, 30
Offset: 0

Views

Author

Gary W. Adamson, Feb 04 2005

Keywords

Examples

			3rd row (1, 9, 24, 20) with alternating signs = characteristic polynomial 3 X 3 matrix [3 1 1 / 1 3 1 / 1 1 3], x^3 - 9x^2 + 24x - 20.

Crossrefs

Row sums are A006234: 1, 4, 15, 54, 189... Rightmost terms in each row = A001792: 1, 3, 8, 20, 48, 112, 256...(row sums of A103406, the analogous triangle with all 2's in the generating matrix.)
See A103247 for another version.

Extensions

Extended and edited by John W. Layman, Mar 17 2005

A136159 A Chebyshev polynomial triangle of the first kind defined by T(n+1,x) = 3x*T(n,x) - T(n-1,x).

Original entry on oeis.org

1, 1, 3, -1, 9, -4, 27, -15, 1, 81, -54, 7, 243, -189, 36, -1, 729, -648, 162, -10, 2187, -2187, 675, -66, 1, 6561, -7290, 2673, -360, 13, 19683, -24057, 10206, -1755, 105, -1, 59049, -78732, 37908, -7938, 675, -16
Offset: 0

Views

Author

Gary W. Adamson, Dec 16 2007

Keywords

Comments

Row sums (unsigned) give A003688, (starting 1, 1, 4, 13, 43, 142, 469, ...).

Examples

			First few rows of the polynomials are:
1;
x;
3x^2 - 1;
9x^3 - 4x;
27x^4 - 15x^2 + 1;
81x^5 - 54x^3 + 7x;
243x^6 - 189x^4 + 36x^2 - 1;
729x^7 - 648x^5 + 162x^3 - 10x;
...

Crossrefs

Cf. A136158, A003688.
Cf. A000244, A006234, A080419, A080420, A080421, A080422, A080423. [Philippe Deléham, Sep 12 2009]

Programs

  • PARI
    P(n) = if (n==0, 1, if (n==1, x, 3*x*P(n-1) - P(n-2)));
row(n) = select(x->x!=0, Vec(P(n))); \\ Michel Marcus, Apr 15 2018

Formula

T(0,x) = 1, T(1,x) = x, T(n+1,x) = 3x*T(n,x) - T(n-1,x).
G.f: (l - tx)/(1 - 3tx + t^2).
Given triangle A136158, shift down columns to allow for (1, 1, 2, 2, 3, 3, ...) terms in each row.

Extensions

Corrected and extended by Philippe Deléham, Sep 12 2009
Keyword tabf set by Michel Marcus, Apr 15 2018

A231351 a(n) = A231349(n+1)/2.

Original entry on oeis.org

1, 2, 2, 2, 4, 5, 4, 2, 4, 6, 8, 5, 10, 11, 8, 2, 4, 6, 8, 6, 12, 14, 16, 5, 10, 14, 20, 11, 22, 23, 16, 2, 4, 6, 8, 6, 12, 14, 16, 6, 12, 16, 24, 14, 28, 30, 32, 5, 10, 14, 20, 14, 28, 32, 40, 11, 22, 30, 44, 23, 46, 47, 32, 2, 4, 6, 8, 6, 12, 14, 16
Offset: 1

Views

Author

Omar E. Pol, Dec 17 2013

Keywords

Comments

Observation: the row sums of the first six rows coincide with the first six elements of A006234.
Is A006234 the row sums of this triangle?

Examples

			Written as an irregular triangle in which row lengths is A000079 the sequence begins:
1;
2,2;
2,4,5,4;
2,4,6,8,5,10,11,8;
2,4,6,8,6,12,14,16,5,10,14,20,11,22,23,16;
2,4,6,8,6,12,14,16,6,12,16,24,14,28,30,32,5,10,14,20,14,28,32,40,11,22,30,44,23,46,47,32;

Links

Crossrefs

Right border gives A000079.
Cf. A006234, A231348, A231349.
Previous Showing 31-34 of 34 results.

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

Content is available under The OEIS End-User License Agreement.