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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A342134 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of g.f. 1/(1 - 2*k*x - k*x^2).

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 5, 0, 1, 6, 18, 12, 0, 1, 8, 39, 80, 29, 0, 1, 10, 68, 252, 356, 70, 0, 1, 12, 105, 576, 1629, 1584, 169, 0, 1, 14, 150, 1100, 4880, 10530, 7048, 408, 0, 1, 16, 203, 1872, 11525, 41344, 68067, 31360, 985, 0, 1, 18, 264, 2940, 23364, 120750, 350272, 439992, 139536, 2378, 0
Offset: 0

Views

Author

Seiichi Manyama, Mar 01 2021

Keywords

Examples

			Square array begins:
  1,  1,    1,     1,     1,      1, ...
  0,  2,    4,     6,     8,     10, ...
  0,  5,   18,    39,    68,    105, ...
  0, 12,   80,   252,   576,   1100, ...
  0, 29,  356,  1629,  4880,  11525, ...
  0, 70, 1584, 10530, 41344, 120750, ...

Links

Crossrefs

Columns 0..5 give A000007, A000129(n+1), A090017(n+1), A090018, A190510(n+1), A190955(n+1).
Rows 0..2 give A000012, A005843, A007742.
Main diagonal gives A109517(n+1).
Cf. A342120, A342129, A342133.

Programs

  • Maple
    T:= (n, k)-> (<<0|1>, >^(n+1))[1, 2]:
seq(seq(T(n, d-n), n=0..d), d=0..12); # Alois P. Heinz, Mar 01 2021
  • Mathematica
    T[n_, k_] := Sum[If[k == j == 0, 1, (2*k)^j] * 2^(j - n) * Binomial[j, n - j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Apr 27 2021 *)
  • PARI
    T(n, k) = sum(j=0, n\2, (2*k)^(n-j)*2^(-j)*binomial(n-j, j));
  • PARI
    T(n, k) = sum(j=0, n, (2*k)^j*2^(j-n)*binomial(j, n-j));
  • PARI
    T(n, k) = round((-sqrt(k)*I)^n*polchebyshev(n, 2, sqrt(k)*I));

Formula

T(0,k) = 1, T(1,k) = 2*k and T(n,k) = k*(2*T(n-1,k) + T(n-2,k)) for n > 1.
T(n,k) = Sum_{j=0..floor(n/2)} (2*k)^(n-j) * (1/2)^j * binomial(n-j,j) = Sum_{j=0..n} (2*k)^j * (1/2)^(n-j) * binomial(j,n-j).
T(n,k) = (-sqrt(k)*i)^n * U(n, sqrt(k)*i) where U(n, x) is a Chebyshev polynomial of the second kind.

A109519 a(n) is the (1,2)-entry of the n-th power of the 2 X 2 matrix [0,-1;n-1,n-1].

Original entry on oeis.org

-1, -1, -2, -9, -80, -1000, -15336, -276115, -5705728, -133155495, -3464900000, -99490865760, -3125217447936, -106614813012877, -3925516139359360, -155164259295703125, -6553564019985219584, -294562012662334323872, -14038370700094085018112
Offset: 1

Views

Author

Roger L. Bagula, Jun 16 2005

Keywords

Comments

The (1,2)-entry of the n-th power of the 2 X 2 matrix [0,1;1,1] is the Fibonacci number A000045(n).

Examples

			a(4)=-9 because if M is the 2 X 2 matrix [0,-1;3,3], then M^4 is the 2 X 2 matrix [ -18,-9,27,9].

Crossrefs

Cf. A000045, A000166, A109516, A109517, A109518.

Programs

  • Maple
    with(linalg): a:=proc(n) local A,k: A[1]:=matrix(2,2,[0,-1,n-1,n-1]): for k from 2 to n do A[k]:=multiply(A[k-1],A[1]) od: A[n][1,2] end: seq(a(n),n=1..21);
  • Mathematica
    M[n_] = If[n > 1, MatrixPower[{{0, -1}, {n - 1, (n - 1)}}, n], {{0, 1}, {1, 1}}] a = Table[Abs[M[n][[1, 2]]], {n, 1, 50}]
  • PARI
    a(n) = round(-sqrt(n-1)^(n-1)*polchebyshev(n-1, 2, sqrt(n-1)/2)); \\ Seiichi Manyama, Feb 28 2021
  • Sage
    [ -lucas_number1(n+1,n,n) for n in range(0,19)] # Zerinvary Lajos, Jul 16 2008

Formula

From Seiichi Manyama, Feb 28 2021: (Start)
a(n+1) = [x^n] 1/(-1 + n*x - n*x^2).
a(n+1) = (-1)^(n+1) * Sum_{k=0..n} (-n)^k * binomial(k,n-k).
a(n+1) = (-1) * sqrt(n)^n * S(n, sqrt(n)) with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind. (End)
Showing 1-2 of 2 results.

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