%I #54 Aug 20 2024 23:19:41

%S 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,

%T 7,37,135,305,360,169,21,1,8,50,228,701,1292,1189,408,34,1,9,65,357,

%U 1405,3640,5473,3927,985,55,1,10,82,528,2549,8658,18901,23184,12970,2378,89

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

%C 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

%C 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

%C As to the array sequences, (n+1)-th sequence is the INVERT transform of the n-th sequence. - _Gary W. Adamson_, Aug 20 2013

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

%C ...

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

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

%C l, 0, 1, 0, 1, 0, ...

%C 1, 1, 2, 3, 5, 8, ...

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

%C ...

%C 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

%C From _Michael A. Allen_, Feb 21 2023: (Start)

%C Row n is the n-metallonacci sequence.

%C 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)

%H G. C. Greubel, <a href="/A073133/b073133.txt">Antidiagonals n = 1..100, flattened</a>

%H Michael A. Allen and Kenneth Edwards, <a href="https://www.fq.math.ca/Papers1/60-5/allen.pdf">Fence tiling derived identities involving the metallonacci numbers squared or cubed</a>, Fib. Q. 60:5 (2022) 5-17.

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

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

%F G.f. for row n: x/(1-n*x-x^2). - _L. Edson Jeffery_, Aug 28 2013

%e Table begins:

%e 1, 1, 2, 3, 5, 8, 13, ...

%e 1, 2, 5, 12, 29, 70, 169, ...

%e 1, 3, 10, 33, 109, 360, 1189, ...

%e 1, 4, 17, 72, 305, 1292, 5473, ... etc.

%p A073133 := proc(n,k)

%p option remember;

%p if k <= 1 then

%p k;

%p else

%p n*procname(n,k-1)+procname(n,k-2) ;

%p end if;

%p end proc:

%p seq(seq( A073133(d-k,k),k=1..d-1),d=2..13) ; # _R. J. Mathar_, Aug 16 2019

%t 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 *)

%o (PARI) T(n,k) = if(k==1, 1, k<0, 0, n*T(n,k-1)+T(n,k-2));

%o for(n=1,15, for(k=1,n, print1(T(n-k+1,k), ", "))) \\ _G. C. Greubel_, Aug 12 2019

%o (Sage)

%o def T(n, k):

%o if (k<0): return 0

%o elif (k==1): return 1

%o else: return n*T(n, k-1) + T(n, k-2)

%o [[T(n-k+1, k) for k in (1..n)] for n in (1..15)] # _G. C. Greubel_, Aug 12 2019

%o (GAP)

%o T:= function(n,k)

%o if k<0 then return 0;

%o elif k=1 then return 1;

%o else return n*T(n,k-1) + T(n,k-2);

%o fi;

%o end;

%o Flat(List([1..15], n-> List([1..n], k-> T(n-k+1,k) ))); # _G. C. Greubel_, Aug 12 2019

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

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

%Y Different from A081572.

%K nonn,tabl,easy

%O 1,5

%A _Henry Bottomley_, Jul 16 2002