A073133 Table by antidiagonals of T(n,k) = n*T(n,k-1) + T(n,k-2) starting with T(n,1) = 1.

1, 1, 1, 1, 2, 2, 1, 3, 5, 3, 1, 4, 10, 12, 5, 1, 5, 17, 33, 29, 8, 1, 6, 26, 72, 109, 70, 13, 1, 7, 37, 135, 305, 360, 169, 21, 1, 8, 50, 228, 701, 1292, 1189, 408, 34, 1, 9, 65, 357, 1405, 3640, 5473, 3927, 985, 55, 1, 10, 82, 528, 2549, 8658, 18901, 23184, 12970, 2378, 89

Offset: 1

Author

Henry Bottomley, Jul 16 2002

Keywords

• nonn
• tabl
• easy

Comments

Columns of the array are generated from Fibonacci polynomials f(x). They are: (1), (x), (x^2 + 1), (x^3 + 2x), (x^4 + 3x^2 + 1), (x^5 + 4x^3 + 3x), (x^6 + 5x^4 + 6x^2 +1), ... If column headings start 0, 1, 2, ... then the terms in the n-th column are generated from the n-th degree Fibonacci polynomial. For example, column 5 (8, 70, 360, ...) is generated from f(x), x = 1,2,3,...; fifth-degree polynomial x^5 + 4x^3 + 3x; e.g., f(2) = 70 = 2^5 + 4*8 + 3*2. - Gary W. Adamson, Apr 02 2006

The ratio of two consecutive entries of the sequence in the n-th row approaches (n + sqrt(n^2 + 4))/2. Example: The sequence beginning (1, 3, 10, 33, ...) tends to 3.302775... = (3 + sqrt(13))/2. - Gary W. Adamson, Aug 12 2013

As to the array sequences, (n+1)-th sequence is the INVERT transform of the n-th sequence. - Gary W. Adamson, Aug 20 2013

The array can be extended infinitely above the Fibonacci row by taking successive INVERTi transforms, resulting in:

...

1, -2, 5, -12, 29, -70, ...

1, -1, 2, -3, 5, -8, ...

l, 0, 1, 0, 1, 0, ...

1, 1, 2, 3, 5, 8, ...

1, 2, 5, 12, 29, 70, ...

...

This results in an infinite array in which sequences above the (1, 0, 1, 0, ...) are reflections of the sequences below, except for the alternate signs. Any sequence in the (+ sign) row starting (1, n, ...) is the (2*n-th) INVERT transform of the same sequence but with alternate signs. Example: (1, 2, 5, 12, ...) is the (2*2) = fourth INVERT transform of (1, -2, 5, -12, ...) by inspection. Conjecture: This "reflection" principle will result from taking successive INVERT transforms of any aerated sequence starting 1, ... and with positive signs. Likewise, the rows above the aerated sequence are successive INVERTi transforms of the aerated sequence. - Gary W. Adamson, Jul 14 2019

From Michael A. Allen, Feb 21 2023: (Start)

Row n is the n-metallonacci sequence.

T(n,k) is the number of tilings of a (k-1)-board (a board with dimensions (k-1) X 1) using unit squares and dominoes (with dimensions 2 X 1) if there are n kinds of squares available. (End)

Examples

			Table begins:
  1, 1,  2,  3,   5,    8,   13, ...
  1, 2,  5, 12,  29,   70,  169, ...
  1, 3, 10, 33, 109,  360, 1189, ...
  1, 4, 17, 72, 305, 1292, 5473, ... etc.
		

Links

• G. C. Greubel, Antidiagonals n = 1..100, flattened
• Michael A. Allen and Kenneth Edwards, Fence tiling derived identities involving the metallonacci numbers squared or cubed, Fib. Q. 60:5 (2022) 5-17.

Crossrefs

Rows are A000045, A000129, A006190, A001076, A052918, A005668, A054413, A041025, A099371, A041041, A049666, A041061, A140455, A041085, etc.

Columns are A000012, A000027, A002522, A054602, A057721, A124152, etc.

Different from A081572.

Programs

GAP

T:= function(n,k)
    if k<0 then return 0;
    elif k=1 then return 1;
    else return n*T(n,k-1) + T(n,k-2);
    fi;
  end;
Flat(List([1..15], n-> List([1..n], k-> T(n-k+1,k) ))); # G. C. Greubel, Aug 12 2019

Maple

A073133 := proc(n,k)
    option remember;
    if k <= 1 then
        k;
    else
        n*procname(n,k-1)+procname(n,k-2) ;
    end if;
end proc:
seq(seq( A073133(d-k,k),k=1..d-1),d=2..13) ; # R. J. Mathar, Aug 16 2019

Mathematica

T[n_, 1]:= 1; T[n_, k_]:= T[n, k] = If[k<0, 0, n*T[n, k-1] + T[n, k-2]]; Table[T[n-k+1, k], {n, 15}, {k, n}]//Flatten (* G. C. Greubel, Aug 12 2019 *)

PARI

T(n,k) = if(k==1, 1, k<0, 0, n*T(n,k-1)+T(n,k-2));
for(n=1,15, for(k=1,n, print1(T(n-k+1,k), ", "))) \\ G. C. Greubel, Aug 12 2019

Sage

def T(n, k):
    if (k<0): return 0
    elif (k==1): return 1
    else: return n*T(n, k-1) + T(n, k-2)
[[T(n-k+1, k) for k in (1..n)] for n in (1..15)] # G. C. Greubel, Aug 12 2019

Formula

T(n, k) = A073134(n, k) + 2*A073135(n, k-2) = Sum_{j=0..k-1} abs(A049310(k-1, j)*n^j).

T(n,k) = [[0,1; 1,n]^{k+1}]{1,1}, n,k in {1,2,...}. - _L. Edson Jeffery, Sep 23 2012

G.f. for row n: x/(1-n*x-x^2). - L. Edson Jeffery, Aug 28 2013