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 1-2 of 2 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)

A140870 8*P_4(2n), 8 times the Legendre Polynomial of order 4 at 2n.

Original entry on oeis.org

3, 443, 8483, 44283, 141443, 347003, 721443, 1338683, 2286083, 3664443, 5588003, 8184443, 11594883, 15973883, 21489443, 28323003, 36669443, 46737083, 58747683, 72936443, 89552003, 108856443, 131125283, 156647483, 185725443, 218675003, 255825443
Offset: 0

Views

Author

N. J. A. Sloane, Nov 17 2009

Keywords

Links

Crossrefs

Cf. A144124.

Programs

  • Magma
    P := PolynomialRing(IntegerRing());
LP4:=LegendrePolynomial(4);
[ Evaluate(8*LP4, 2*n): n in [0..26] ]; // Klaus Brockhaus, Nov 18 2009
  • Magma
    [560*n^4 - 120*n^2 + 3: n in [0..30]]; // Vincenzo Librandi, Oct 04 2015
  • Maple
    A140870 := proc(n)
        8*orthopoly[P](4,2*n) ;
end proc: # R. J. Mathar, Oct 24 2011
  • Mathematica
    Table[8 LegendreP[4,2n],{n,0,50}]
LinearRecurrence[{5, -10, 10, -5, 1}, {3, 443, 8483, 44283, 141443}, 30] (* Vincenzo Librandi, Oct 04 2015 *)
  • PARI
    {for(n=0, 26, print1(subst(8*pollegendre(4), x, 2*n), ","))} \\ Klaus Brockhaus, Nov 21 2009

Formula

Legendre polynomial LP_4(x) = (35*x^4-30*x^2+3)/8. - Klaus Brockhaus, Nov 21 2009
From Klaus Brockhaus, Nov 21 2009: (Start)
a(n) = 560*n^4-120*n^2+3.
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4)+13440 for n > 3; a(0)=3, a(1)=443, a(2)=8483, a(3)=44283.
G.f.: (3+428*x+6298*x^2+6268*x^3+443*x^4)/(1-x)^5. (End)
Showing 1-2 of 2 results.

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

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