A109519 -id:A109519 - cp's OEIS frontend

A109516 a(n) is the (1,2)-entry of the n-th power of the 2 X 2 matrix [0,1;n-1,n-1].

1, 1, 6, 45, 464, 6000, 93528, 1707111, 35721216, 843160671, 22165100000, 642268811184, 20339749638144, 698946255836933, 25903663544572800, 1029945249481640625, 43733528272753917952, 1975222567881226040760

Offset: 1

Roger L. Bagula, Jun 16 2005

nonn

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).

			a(4)=45 because if M is the 2 X 2 matrix [0,1;3,3], then M^4 is the 2 X 2 matrix [36,45;135;171].
G.f. = x + x^2 + 6*x^3 + 45*x^4 + 464*x^5 + 6000*x^6 + 93528*x^7 + 1707111*x^8 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..387

Cf. A000045, A000166, A109519.

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..20);

M[n_] = If[n > 1, MatrixPower[{{0, 1}, {n - 1, n - 1}}, n], {{0, 1}, {1, 1}}]; a = Table[M[n][[1, 2]], {n, 1, 50}]
Table[SeriesCoefficient[1/(1 - n*x - n*x^2), {x,0,n}], {n,0,20}] (* Vaclav Kotesovec, Apr 20 2018 *)

{a(n)=polcoeff(1/(1-n*x-n*x^2+x*O(x^n)), n)} \\ Paul D. Hanna, Dec 27 2012

a(n) = ([0,1;n-1,n-1]^n)[1, 2]; \\ Michel Marcus, Apr 20 2018

a(n) = round((-sqrt(n-1)*I)^(n-1)*polchebyshev(n-1, 2, sqrt(n-1)*I/2)); \\ Seiichi Manyama, Feb 28 2021

a(n+1) = [x^n] 1/(1 - n*x - n*x^2). - Paul D. Hanna, Dec 27 2012
a(n+1) = Sum_{k=0..floor(n/2)} binomial(n-k,k)*n^(n-k) for n>=0 (conjectured). - Werner Schulte, Oct 21 2016
a(n) = ((n + sqrt((n-1)*(n+3)) - 1)^n - (n - sqrt((n-1)*(n+3)) - 1)^n) / (2^n * sqrt((n-1)*(n+3))), for n > 1. - Daniel Suteu, Apr 20 2018
a(n) ~ n^(n-1). - Vaclav Kotesovec, Apr 20 2018
a(n+1) = (-sqrt(n)*i)^n * S(n, sqrt(n)*i) with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind. - Seiichi Manyama, Feb 28 2021

A342129 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of g.f. 1/(1 - kx + kx^2).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 2, -1, 0, 1, 4, 6, 0, -1, 0, 1, 5, 12, 9, -4, 0, 0, 1, 6, 20, 32, 9, -8, 1, 0, 1, 7, 30, 75, 80, 0, -8, 1, 0, 1, 8, 42, 144, 275, 192, -27, 0, 0, 0, 1, 9, 56, 245, 684, 1000, 448, -81, 16, -1, 0, 1, 10, 72, 384, 1421, 3240, 3625, 1024, -162, 32, -1, 0

Offset: 0

Seiichi Manyama, Feb 28 2021

			Square array begins:
  1,  1,  1, 1,   1,    1, ...
  0,  1,  2, 3,   4,    5, ...
  0,  0,  2, 6,  12,   20, ...
  0, -1,  0, 9,  32,   75, ...
  0, -1, -4, 9,  80,  275, ...
  0,  0, -8, 0, 192, 1000, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Index entries for sequences related to Chebyshev polynomials.

Columns 0..10 give A000007, A010892, A099087, A057083, A001787(n+1), A030191, A030192, A030240, A057084, A057085(n+1), A057086.
Rows 0..1 give A000012, A001477.
Main diagonal gives (-1) * A109519(n+1).
Cf. A342120, A342133, A342134.

T:= (n, k)-> (<<0|1>, <-k|k>>^(n+1))[1, 2]:
seq(seq(T(n, d-n), n=0..d), d=0..12); # Alois P. Heinz, Mar 01 2021

T[n_, k_] := (-1)^n * Sum[If[k == j == 0, 1, (-k)^j] * Binomial[j, n - j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, Apr 28 2021 *)

T(n, k) = (-1)^n*sum(j=0, n\2, (-k)^(n-j)*binomial(n-j, j));

T(n, k) = (-1)^n*sum(j=0, n, (-k)^j*binomial(j, n-j));

T(n, k) = round(sqrt(k)^n*polchebyshev(n, 2, sqrt(k)/2));

T(0,k) = 1, T(1,k) = k and T(n,k) = k*(T(n-1,k) - T(n-2,k)) for n > 1.
T(n,k) = (-1)^n * Sum_{j=0..floor(n/2)} (-k)^(n-j) * binomial(n-j,j) = (-1)^n * Sum_{j=0..n} (-k)^j * binomial(j,n-j).
T(n,k) = sqrt(k)^n * S(n, sqrt(k)) with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind.

cp's OEIS Frontend

A109516 a(n) is the (1,2)-entry of the n-th power of the 2 X 2 matrix [0,1;n-1,n-1].

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A342129 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of g.f. 1/(1 - kx + kx^2).

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cp's OEIS Frontend

A109516 a(n) is the (1,2)-entry of the n-th power of the 2 X 2 matrix [0,1;n-1,n-1].

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

Formula

A342129 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of g.f. 1/(1 - k*x + k*x^2).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

Formula

A342129 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of g.f. 1/(1 - kx + kx^2).