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.

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A100259 Coefficient of x^2 in 2n-th normalized Legendre polynomial.

Original entry on oeis.org

3, -30, 105, -1260, 3465, -18018, 45045, -875160, 2078505, -9699690, 22309287, -202811700, 456326325, -2035917450, 4508102925, -158685222960, 347123925225, -1511010027450, 3273855059475, -28258538408100
Offset: 1

Views

Author

Ralf Stephan, Nov 13 2004

Keywords

Examples

			Example: Legendre(12) = (676039x^12 - 1939938x^10 + 2078505x^8 - 1021020x^6 + 225225x^4 - 18018x^2 + 231)/1024, hence a(6) = -18018. - Neven Juric

Crossrefs

3rd column (without zeros) of triangle A100258.

Programs

  • PARI
    a(n) = polcoeff(pollegendre(2*n, x), 2)*2^valuation((2*n)!, 2) \\ Michel Marcus, May 29 2013

Extensions

Definition corrected by Neven Juric, Jan 10 2009

A356205 T(n,k) are the numerators of the coefficients of the Legendre polynomials of degree n, with increasing exponents, where T(n,k) is a triangle read by rows.

Original entry on oeis.org

1, 0, 1, -1, 0, 3, 0, -3, 0, 5, 3, 0, -15, 0, 35, 0, 15, 0, -35, 0, 63, -5, 0, 105, 0, -315, 0, 231, 0, -35, 0, 315, 0, -693, 0, 429, 35, 0, -315, 0, 3465, 0, -3003, 0, 6435, 0, 315, 0, -1155, 0, 9009, 0, -6435, 0, 12155, -63, 0, 3465, 0, -15015, 0, 45045, 0, -109395, 0, 46189
Offset: 0

Views

Author

Hugo Pfoertner, Jul 29 2022

Keywords

Examples

			The triangle begins:
   1;
   0,   1;
  -1,   0,    3;
   0,  -3,    0,     5;
   3,   0,  -15,     0,   35;
   0,  15,    0,   -35,    0,   63;
  -5,   0,  105,     0, -315,    0,   231;
   0, -35,    0,   315,    0, -693,     0,   429;
  35,   0, -315,     0, 3465,    0, -3003,     0, 6435;
   0, 315,    0, -1155,    0, 9009,     0, -6435,    0, 12155
.
Fractions:
   \ k 0        1       2      3       4       5      6       7      8
  n \ -------------------------------------------------------------------
  0 |  1        .       .      .       .       .       .      .      .
  1 |  0       1        .      .       .       .       .      .      .
  2 | -1/2     0       3/2     .       .       .       .      .      .
  3 |  0      -3/2     0      5/2      .       .       .      .      .
  4 |  3/8     0     -15/4    0      35/8      .       .      .      .
  5 |  0      15/8     0    -35/4     0      63/8      .      .      .
  6 | -5/16    0     105/16   0    -315/16    0     231/16    .      .
  7 |  0     -35/16    0    315/16    0    -693/16    0    429/16    .
  8 | 35/128   0    -315/32   0    3465/64    0   -3003/32   0   6435/128

Crossrefs

A356206 are the corresponding denominators.
Cf. A005187, A100258.

Programs

  • PARI
    for (n=0, 10, my(P=pollegendre(n,'x));for (j=0, n, print1(numerator(polcoef(P,j)),", ")); print())

A157077 Triangle read by rows, coefficients of the Legendre polynomials P(n, x) times 2^n: T(n, k) = 2^n * [x^k] P(n, x), n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 0, 2, -2, 0, 6, 0, -12, 0, 20, 6, 0, -60, 0, 70, 0, 60, 0, -280, 0, 252, -20, 0, 420, 0, -1260, 0, 924, 0, -280, 0, 2520, 0, -5544, 0, 3432, 70, 0, -2520, 0, 13860, 0, -24024, 0, 12870, 0, 1260, 0, -18480, 0, 72072, 0, -102960, 0, 48620, -252, 0, 13860, 0, -120120, 0, 360360, 0, -437580, 0, 184756
Offset: 0

Views

Author

Roger L. Bagula, Feb 22 2009

Keywords

Examples

			The term order is Q(x) = a_0 + a_1*x + ... + a_n*x^n. The coefficients of the first few polynomials in this order are:
{1},
{0, 2},
{-2, 0, 6},
{0, -12, 0, 20},
{6, 0, -60, 0, 70},
{0, 60, 0, -280, 0, 252},
{-20, 0, 420, 0, -1260, 0, 924},
{0, -280, 0, 2520, 0, -5544, 0, 3432},
{70, 0, -2520, 0, 13860, 0, -24024, 0, 12870},
{0, 1260, 0, -18480, 0, 72072, 0, -102960, 0, 48620},
{-252, 0, 13860, 0, -120120, 0, 360360, 0, -437580, 0, 184756}.
.
From _Jon E. Schoenfield_, Jul 04 2022: (Start)
As a right-aligned triangle:
                                                             1;
                                                     0,      2;
                                                 -2, 0,      6;
                                         0,     -12, 0,     20;
                                      6, 0,     -60, 0,     70;
                              0,     60, 0,    -280, 0,    252;
                         -20, 0,    420, 0,   -1260, 0,    924;
                  0,    -280, 0,   2520, 0,   -5544, 0,   3432;
              70, 0,   -2520, 0,  13860, 0,  -24024, 0,  12870;
        0,  1260, 0,  -18480, 0,  72072, 0, -102960, 0,  48620;
  -252, 0, 13860, 0, -120120, 0, 360360, 0, -437580, 0, 184756. (End)

Links

Crossrefs

Cf. A100258, A126869, A000984, A005430, A000079, A069835, A084773, A098269, A098270.

Programs

  • Maple
    with(orthopoly):with(PolynomialTools): seq(print(CoefficientList (2^n*P(n, x), x,termorder=forward)),n=0..10); # Peter Luschny, Dec 18 2014
  • Mathematica
    Table[CoefficientList[2^n*LegendreP[n, x], x], {n, 0, 10}]; Flatten[%]
  • PARI
    tabl(nn) = for (n=0, nn, print(Vecrev(2^n*pollegendre(n)))); \\ Michel Marcus, Dec 18 2014
  • Sage
    def A157077_row(n):
    if n==0: return [1]
    T = [c[0] for c in (2^n*gen_legendre_P(n, 0, x)).coefficients()]
    return [0 if is_odd(n+k) else T[k//2] for k in (0..n)]
for n in range(9): print(A157077_row(n)) # Peter Luschny, Dec 19 2014

Formula

Row sums are 2^n.
From Peter Luschny, Dec 19 2014: (Start)
T(n,0) = A126869(n).
T(n,n) = A000984(n).
T(n,1) = (-1)^floor(n/2)*A005430(floor(n/2)+1) if n is odd else 0.
Let Q(n, x) = 2^n*P(n, x).
Q(n,0) = (-1)^floor(n/2)*A126869(floor(n/2)) if n is even else 0.
Q(n,1) = A000079(n).
Q(n,2) = A069835(n).
Q(n,3) = A084773(n).
Q(n,4) = A098269(n).
Q(n,5) = A098270(n). (End)
From Fabián Pereyra, Jun 30 2022: (Start)
n*T(n,k) = 2*(2*n-1)*T(n-1,k-1) - 4*(n-1)*T(n-2,k).
T(n,k) = (-1)^floor((n-k)/2)*binomial(n+k,k)*binomial(n,floor((n-k)/2))*(1+(-1)^(n-k))/2.
O.g.f.: A(x,t) = 1/sqrt(1-4*x*t+4*x^2) = 1 + (2*t)*x + (-2+6*t^2)*x^2 + (-12*t+20*t^3)*x^3 + (6-60*t^2+70*t^4)*x^4 + .... (End)

Extensions

Name clarified and edited by Peter Luschny, Dec 18 2014

A245320 Coefficients of "optimum L" polynomials L_n(ω^2) ordered by increasing powers.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 1, -3, 3, 0, 0, 3, -8, 6, 0, 1, -8, 28, -40, 20, 0, 0, 6, -40, 105, -120, 50, 0, 1, -15, 105, -355, 615, -525, 175, 0, 0, 10, -120, 615, -1624, 2310, -1680, 490, 0, 1, -24, 276, -1624, 5376, -10416, 11704, -7056, 1764, 0, 0, 15, -280
Offset: 0

Views

Author

Jonathan Bright, Jul 17 2014

Keywords

Comments

Used in the generation of "optimum L" (or Legendre-Papoulis) filters.

Examples

			Triangle begins:
0;
0, 1;
0, 0,  1;
0, 1, -3,   3;
0, 0,  3,  -8,  6;
0, 1, -8,  28, -40,   20;
0, 0,  6, -40, 105, -120, 50;
...
So:
L_4(ω^2) = 0 + 0ω^2 + 3ω^4 -  8ω^6 +  6ω^8
L_5(ω^2) = 0 + 1ω^2 - 8ω^4 + 28ω^6 - 40ω^8 + 20ω^10

References

  • A. Papoulis, ”On Monotonic Response Filters,” Proc. IRE, 47, No. 2, Feb. 1959, 332-333 (correspondence section)

Links

Crossrefs

Derived from A100258 and A060818.
Previous Showing 21-24 of 24 results.

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