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.

A128496 Row sums of unsigned triangle |A128495|=|S(2;n,m)| (sums of squares of Chebyshev's S-polynomials).

Original entry on oeis.org

1, 2, 4, 9, 20, 50, 125, 324, 840, 2195, 5736, 15012, 39289, 102854, 269260, 704925, 1845500, 4831574, 12649205, 33116040, 86698896, 226980647, 594243024, 1555748424, 4073002225, 10663258250, 27916772500, 73087059249, 191344405220
Offset: 0

Views

Author

Wolfdieter Lang Apr 04 2007

Keywords

Crossrefs

Cf. A004523 (looks like signed row sums).

Formula

a(n)=sum(|A128495(n,m)|,m=0..n).

A217476 Coefficient triangle for the square of the monic integer Chebyshev T-polynomials A127672.

Original entry on oeis.org

4, 0, 1, 4, -4, 1, 0, 9, -6, 1, 4, -16, 20, -8, 1, 0, 25, -50, 35, -10, 1, 4, -36, 105, -112, 54, -12, 1, 0, 49, -196, 294, -210, 77, -14, 1, 4, -64, 336, -672, 660, -352, 104, -16, 1, 0, 81, -540, 1386, -1782, 1287, -546, 135, -18, 1, 4, -100, 825, -2640, 4290, -4004, 2275, -800, 170, -20, 1
Offset: 0

Views

Author

Wolfdieter Lang, Oct 17 2012

Keywords

Comments

The monic integer T-polynomials, called R(n,x) (in Abramowitz-Stegun C(n,x)), with their coefficient triangle given in A127672, when squared, become polynomials in y=x^2:
R(n,x)^2 = sum(T(n,k)*y^k,m=0..n).
R(n,x)^2 = 2 + R(2*n,x). From the bisection of the R-(or T-)polynomials, the even part. Directly from the R(m*n,x)=R(m,R(n,x)) property for m=2.
The o.g.f. is G(z,y) := sum((R(n,sqrt(y))^2)*z^n ,n=0..infinity) = (4 + (4 - 3*y)*z + y*z^2)/((1 +(2-y)*z + z^2)*(1-z)). From the bisection.
The o.g.f.s of the columns k>=1 are x^k*(1-x)/(1+x)^(2*k+1),
and for k=0 the o.g.f. is 4/(1-x^2).
Hetmaniok et al. (2015) refer to these as "modified Chebyshev" polynomials. - N. J. A. Sloane, Sep 13 2016

Examples

			The triangle begins:
n\k 0    1    2      3     4      5     6     7    8   9  10
0:  4
1:  0    1
2:  4   -4    1
3:  0    9   -6      1
4:  4  -16   20     -8     1
5:  0   25  -50     35   -10      1
6:  4  -36  105   -112    54    -12     1
7:  0   49 -196    294  -210     77   -14     1
8:  4  -64  336   -672   660   -352   104   -16    1
9:  0   81 -540   1386 -1782   1287  -546   135  -18   1
10: 4 -100  825  -2640  4290  -4004  2275  -800  170 -20   1
...
n=2:  R(2,x) = -2 + y, R(2,x)^2 = 4 -4*y + y^2, with y=x^2.
n=3:  R(3,x) = 3*x - x^3, R(3,x)^2 = 9*y - 6*y^2 +y^3, with y=x^2.
T(4,1) = 8*(-1)^3*binomial(5,3)/5 = -16.
T(4,0) = 2 + 8*(-1)^4*binomial(4,4)/4 = 4.
T(n,1) = (-1)^(n-1)*2*n*(n+1)!/((n-1)!*2!*(n+1)) = -((-1)^n)*n^2 = A162395(n), n >= 1.
T(n,2) = (-1)^n*A002415(n), n >= 0.
T(n,3) = -(-1)^n*A040977(n-3), n >= 3.
T(n,4) = (-1)^n*A053347(n-4), n >= 4.
T(n,5) = -(-1)^n*A054334(n-5), n >= 5.

References

  • E Hetmaniok, P Lorenc, S Damian, et al., Periodic orbits of boundary logistic map and new kind of modified Chebyshev polynomials in R. Witula, D. Slota, W. Holubowski (eds.), Monograph on the Occasion of 100th Birthday Anniversary of Zygmunt Zahorski. Wydawnictwo Politechniki Slaskiej, Gliwice 2015, pp. 325-343.

Crossrefs

Cf. A127672, A158454 (square of S-polynomials), A128495 (sum of square of S-polynomials).

Formula

T(n,k) = [x^(2*k)]R(n,x)^2, with R(n,x) the monic integer version of the Chebyshev T(n,x) polynomial.
T(n,k) = 0 if n=1. ([k=0] means 1 if k=0 else 0).

A128494 Coefficient table for sums of Chebyshev's S-Polynomials.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, -1, 1, 1, 1, -1, -2, 1, 1, 1, 2, -2, -3, 1, 1, 0, 2, 4, -3, -4, 1, 1, 0, -2, 4, 7, -4, -5, 1, 1, 1, -2, -6, 7, 11, -5, -6, 1, 1, 1, 3, -6, -13, 11, 16, -6, -7, 1, 1, 0, 3, 9, -13, -24, 16, 22, -7, -8, 1, 1, 0, -3, 9, 22, -24, -40, 22, 29, -8, -9, 1, 1, 1, -3, -12, 22, 46, -40, -62, 29, 37, -9, -10, 1, 1, 1, 4, -12
Offset: 0

Views

Author

Wolfdieter Lang, Apr 04 2007

Keywords

Comments

See A049310 for the coefficient table of Chebyshev's S(n,x)=U(n,x/2) polynomials.
This is a 'repetition triangle' based on a signed version of triangle A059260: a(2*p,2*k) = a(2*p+1,2*k) = A059260(p+k,2*k)*(-1)^(p+k) and a(2*p+1,2*k+1) = a(2*p+2,2*k+1) = A059260(p+k+1,2*k+1)*(-1)^(p+k), k >= 0.

Examples

			The triangle a(n,m) begins:
  n\m  0   1   2   3   4   5   6   7   8   9  10
   0:  1
   1:  1   1
   2:  0   1   1
   3:  0  -1   1   1
   4:  1  -1  -2   1   1
   5:  1   2  -2  -3   1   1
   6:  0   2   4  -3  -4   1   1
   7:  0  -2   4   7  -4  -5   1   1
   8:  1  -2  -6   7  11  -5  -6   1   1
   9:  1   3  -6 -13  11  16  -6  -7   1   1
  10:  0   3   9 -13 -24  16  22  -7  -8   1   1
... reformatted by _Wolfdieter Lang_, Oct 16 2012
Row polynomial S(1;4,x) = 1 - x - 2*x^2 + x^3 + x^4 = Sum_{k=0..4} S(k,x).
S(4,y)*S(5,y)/y = 3 - 13*y^2 + 16*y^4 - 7*y^6 + y^8, with y=sqrt(2+x) this becomes S(1;4,x).
From _Wolfdieter Lang_, Oct 16 2012: (Start)
S(1;4,x) = (1 - (S(5,x) - S(4,x)))/(2-x) = (1-x)*(2-x)*(1+x)*(1-x-x^2)/(2-x) = (1-x)*(1+x)*(1-x-x^2).
S(5,x) - S(4,x) = R(11,sqrt(2+x))/sqrt(2+x) = -1 + 3*x + 3*x^2 - 4*x^3 - x^4 + x^5. (End)

Links

Crossrefs

Row sums (signed): A021823(n+2). Row sums (unsigned): A070550(n).
Cf. A128495 for S(2; n, x) coefficient table.
The column sequences (unsigned) are, for m=0..4: A021923, A002265, A008642, A128498, A128499.
For m >= 1 the column sequences (without leading zeros) are of the form a(m, 2*k) = a(m, 2*k+1) = ((-1)^k)*b(m, k) with the sequences b(m, k), given for m=1..11 by A008619, A002620, A002623, A001752, A001753, A001769, A001779, A001780, A001781, A001786, A001808.

Formula

S(1;n,x) = Sum_{k=0..n} S(k,x) = Sum_{m=0..n} a(n,m)*x^m, n >= 0.
a(n,m) = [x^m](S(n,y)*S(n+1,y)/y) with y:=sqrt(2+x).
G.f. for column m: (x^m)/((1-x)*(1+x^2)^(m+1)), which shows that this is a lower diagonal matrix of the Riordan type, named (1/((1+x^2)*(1-x)), x/(1+x^2)).
From Wolfdieter Lang, Oct 16 2012: (Start)
a(n,m) = [x^m](1- (S(n+1,x) - S(n,x)))/(2-x). From the Binet - de Moivre formula for S and use of the geometric sum.
a(n,m) = [x^m](1- R(2*n+3,sqrt(2+x))/sqrt(2+x))/(2-x) with the monic integer T-polynomials R with coefficient triangle given in A127672. From the odd part of the bisection of the T-polynomials. (End)

A128497 Coefficient table for sums over product of adjacent Chebyshev S-polynomials.

Original entry on oeis.org

1, 0, 1, 2, -2, 1, 0, 5, -4, 1, 3, -8, 12, -6, 1, 0, 14, -28, 23, -8, 1, 4, -20, 58, -68, 38, -10, 1, 0, 30, -108, 171, -136, 57, -12, 1, 5, -40, 188, -382, 405, -240, 80, -14, 1, 0, 55, -308, 781, -1056, 828, -388, 107, -16, 1, 6, -70, 483, -1488, 2488, -2472, 1524, -588, 138, -18, 1, 0, 91, -728, 2678, -5408
Offset: 0

Views

Author

Wolfdieter Lang Apr 04 2007

Keywords

Comments

See A049310 for the coefficient table of Chebyshev's S(n,x)=U(n,x/2) polynomials.
See A128495 for the sums of squares of Chebyshev's S-polynomials.

Examples

			1;
0,1;
2,-2,1;
0,5,-4,1;
3,-8,12,-6,1;
0,14,-28,23,-8,1;
...

Links

Crossrefs

Row sums (signed) look like A008620. Row sums (unsigned) look like |A077916|.

Formula

sum(S(k,x)*S(k+1,x)/x,k=0..n)= sum(a(n,m)*x^(2*m),m=0..n).
Showing 1-4 of 4 results.

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

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