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.

A335333 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2*(2*k+1)*x + x^2).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 13, 1, 1, 7, 37, 63, 1, 1, 9, 73, 305, 321, 1, 1, 11, 121, 847, 2641, 1683, 1, 1, 13, 181, 1809, 10321, 23525, 8989, 1, 1, 15, 253, 3311, 28401, 129367, 213445, 48639, 1, 1, 17, 337, 5473, 63601, 458649, 1651609, 1961825, 265729, 1
Offset: 0

Views

Author

Seiichi Manyama, Jun 02 2020

Keywords

Examples

			Square array begins:
  1,    1,     1,      1,      1,       1, ...
  1,    3,     5,      7,      9,      11, ...
  1,   13,    37,     73,    121,     181, ...
  1,   63,   305,    847,   1809,    3311, ...
  1,  321,  2641,  10321,  28401,   63601, ...
  1, 1683, 23525, 129367, 458649, 1256651, ...

Links

Crossrefs

Columns k=0..4 give A000012, A001850, A006442, A084768, A084769.
Rows n=0..6 give A000012, A005408, A003154(n+1), A160674, A144124, A335338, A144126.
Main diagonal gives A331656.
T(n,n-1) gives A331657.
Cf. A307883, A307884.

Programs

  • Mathematica
    T[n_, k_] := LegendreP[n, 2*k + 1]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, May 03 2021 *)
  • PARI
    T(n, k) = pollegendre(n, 2*k+1);

Formula

T(n,k) is the coefficient of x^n in the expansion of (1 + (2*k+1)*x + k*(k+1)*x^2)^n.
T(n,k) = Sum_{j=0..n} k^j * (k+1)^(n-j) * binomial(n,j)^2.
T(n,k) = Sum_{j=0..n} k^j * binomial(n,j) * binomial(n+j,j).
n * T(n,k) = (2*k+1) * (2*n-1) * T(n-1,k) - (n-1) * T(n-2,k).
T(n,k) = P_n(2*k+1), where P_n is n-th Legendre polynomial.
From Seiichi Manyama, Aug 30 2025: (Start)
T(n,k) = (-1)^n * Sum_{j=0..n} (1/(2*(2*k+1)))^(n-2*j) * binomial(-1/2,j) * binomial(j,n-j).
T(n,k) = Sum_{j=0..floor(n/2)} (k*(k+1))^j * (2*k+1)^(n-2*j) * binomial(n,2*j) * binomial(2*j,j).
E.g.f. of column k: exp((2*k+1)*x) * BesselI(0, 2*sqrt(k*(k+1))*x). (End)

A160739 16*P_6(n), 16 times the Legendre Polynomial of order 6 at n.

Original entry on oeis.org

-5, 16, 10159, 143824, 867211, 3415120, 10373071, 26425744, 59271739, 120704656, 227860495, 404631376, 683245579, 1106013904, 1727242351, 2615311120, 3854919931, 5549499664, 7823790319, 10826585296, 14733641995, 19750758736, 26117017999
Offset: 0

Views

Author

N. J. A. Sloane, Nov 17 2009

Keywords

Links

Crossrefs

Cf. A144126, A160741.

Programs

  • Magma
    [231*n^6 -315*n^4 +105*n^2 -5: n in [0..30]]; // G. C. Greubel, May 02 2018
  • Maple
    A160739 := proc(n)
        16*orthopoly[P](6,n) ;
end proc: # R. J. Mathar, Oct 24 2011
  • Mathematica
    Table[16 LegendreP[6,n],{n,0,40}]
  • PARI
    a(n)=16*pollegendre(6,n) \\ Charles R Greathouse IV, Mar 18 2017
  • PARI
    Vec(-(5 - 51*x - 9942*x^2 - 73222*x^3 - 73047*x^4 - 10047*x^5 - 16*x^6) / (1 - x)^7 + O(x^30)) \\ Colin Barker, Jul 23 2019

Formula

a(n) = 231*n^6 - 315*n^4 + 105*n^2 - 5. - Vaclav Kotesovec, Jul 31 2013
From Colin Barker, Jul 23 2019: (Start)
G.f.: -(5 - 51*x - 9942*x^2 - 73222*x^3 - 73047*x^4 - 10047*x^5 - 16*x^6) / (1 - x)^7.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>6.
(End)

A160741 Numerator of P_6(2n), the Legendre polynomial of order 6 at 2n.

Original entry on oeis.org

-5, 10159, 867211, 10373071, 59271739, 227860495, 683245579, 1727242351, 3854919931, 7823790319, 14733641995, 26117017999, 44040338491, 71215667791, 111123125899, 168143944495, 247704167419, 356428995631, 502307776651, 694869638479, 945369767995
Offset: 0

Views

Author

N. J. A. Sloane, Nov 17 2009

Keywords

Links

Crossrefs

Cf. A160739, A144126.

Programs

  • Maple
    A160741 := proc(n)
        orthopoly[P](6,2*n) ;
        numer(%) ;
end proc: # R. J. Mathar, Oct 24 2011
  • Mathematica
    Table[Numerator[LegendreP[6,2n]],{n,0,40}]
  • PARI
    a(n)=numerator(pollegendre(6,n+n)) \\ Charles R Greathouse IV, Oct 24 2011
  • PARI
    Vec(-(5 - 10194*x - 795993*x^2 - 4516108*x^3 - 4515933*x^4 - 796098*x^5 - 10159*x^6) / (1 - x)^7 + O(x^30)) \\ Colin Barker, Jul 23 2019

Formula

From Colin Barker, Jul 23 2019: (Start)
G.f.: -(5 - 10194*x - 795993*x^2 - 4516108*x^3 - 4515933*x^4 - 796098*x^5 - 10159*x^6) / (1 - x)^7.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>6.
a(n) = -5 + 420*n^2 - 5040*n^4 + 14784*n^6.
(End)
Showing 1-3 of 3 results.

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

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