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A059259 Triangle read by rows giving coefficient T(i,j) of x^i y^j in 1/(1-x-x*y-y^2) = 1/((1+y)(1-x-y)) for (i,j) = (0,0), (1,0), (0,1), (2,0), (1,1), (0,2), ...

%I A059259 #74 Apr 15 2024 13:05:10
%S A059259 1,1,0,1,1,1,1,2,2,0,1,3,4,2,1,1,4,7,6,3,0,1,5,11,13,9,3,1,1,6,16,24,
%T A059259 22,12,4,0,1,7,22,40,46,34,16,4,1,1,8,29,62,86,80,50,20,5,0,1,9,37,91,
%U A059259 148,166,130,70,25,5,1,1,10,46,128,239,314,296,200
%N A059259 Triangle read by rows giving coefficient T(i,j) of x^i y^j in 1/(1-x-x*y-y^2) = 1/((1+y)(1-x-y)) for (i,j) = (0,0), (1,0), (0,1), (2,0), (1,1), (0,2), ...
%C A059259 This sequence provides the general solution to the recurrence a(n) = a(n-1) + k*(k+1)*a(n-2), a(0)=a(1)=1. The solution is (1, 1, k^2 + k + 1, 2*k^2 + 2*k + 1, ...) whose coefficients can be read from the rows of the triangle. The row sums of the triangle are given by the case k=1. These are the Jacobsthal numbers, A001045. Viewed as a square array, its first row is (1,0,1,0,1,...) with e.g.f. cosh(x), g.f. 1/(1-x^2) and subsequent rows are successive partial sums given by 1/((1-x)^n * (1-x^2)). - _Paul Barry_, Mar 17 2003
%C A059259 Conjecture: every second column of this triangle is identical to a column in the square array A071921. For example, column 4 of A059259 (1, 3, 9, 22, 46, ...) appears to be the same as column 3 of A071921; column 6 of A059259 (1, 4, 16, 50, 130, 296, ...) appears to be the same as column 4 of A071921; and in general column 2k of A059259 appears to be the same as column k+1 of A071921. Furthermore, since A225010 is a transposition of A071921 (ignoring the latter's top row and two leftmost columns), there appears to be a correspondence between column 2k of A059259 and row k of A225010. - _Mathew Englander_, May 17 2014
%C A059259 T(n,k) is the number of n-tilings of a (one-dimensional) board that use k (1,1)-fence tiles and n-k squares. A (1,1)-fence is a tile composed of two pieces of width 1 separated by a gap of width 1. - _Michael A. Allen_, Jun 25 2020
%C A059259 See the Edwards-Allen 2020 paper, page 14, for proof of Englander's conjecture. - _Michael De Vlieger_, Dec 10 2020
%H A059259 G. C. Greubel, <a href="/A059259/b059259.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A059259 Joseph Briggs, Alex Parker, Coy Schwieder, and Chris Wells, <a href="https://arxiv.org/abs/2404.07285">Frogs, hats and common subsequences</a>, arXiv:2404.07285 [math.CO], 2024. See p. 28.
%H A059259 Kenneth Edwards and Michael A. Allen, <a href="https://arxiv.org/abs/2009.04649">New Combinatorial Interpretations of the Fibonacci Numbers Squared, Golden Rectangle Numbers, and Jacobsthal Numbers Using Two Types of Tile</a>, arXiv:2009.04649 [math.CO], 2020.
%F A059259 G.f.: 1/(1 - x - x*y - y^2).
%F A059259 As a square array read by antidiagonals, this is T(n, k) = Sum_{i=0..n} (-1)^(n-i)*C(i+k, k). - _Paul Barry_, Jul 01 2003
%F A059259 T(2*n,n) = A026641(n). - _Philippe Deléham_, Mar 08 2007
%F A059259 T(n,k) = T(n-1,k) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = T(2,1) = T(2,2)=1, T(1,1)=0, T(n,k)=0 if k < 0 or if k > n. - _Philippe Deléham_, Nov 24 2013
%F A059259 T(n,0) = 1, T(n,n) = (1+(-1)^n)/2, and T(n,k) = T(n-1,k) + T(n-1,k-1) for 0 < k < n. - _Mathew Englander_, May 24 2014
%F A059259 From _Michael A. Allen_, Jun 25 2020: (Start)
%F A059259 T(n,k) + T(n-1,k-1) = binomial(n,k) if n >= k > 0.
%F A059259 T(2*n-1,2*n-2) = T(2*n,2*n-1) = n, T(2*n,2*n-2) = n^2, T(2*n+1,2*n-1) = n*(n+1) for n > 0.
%F A059259 T(n,2) = binomial(n-2,2) + n - 1 for n > 1 and T(n,3) = binomial(n-3,3) + 2*binomial(n-2,2) for n > 2.
%F A059259 T(2*n-k,k) = A123521(n,k). (End)
%e A059259 Triangle begins:
%e A059259   1;
%e A059259   1,  0;
%e A059259   1,  1,  1;
%e A059259   1,  2,  2,   0;
%e A059259   1,  3,  4,   2,   1;
%e A059259   1,  4,  7,   6,   3,   0;
%e A059259   1,  5, 11,  13,   9,   3,   1;
%e A059259   1,  6, 16,  24,  22,  12,   4,   0;
%e A059259   1,  7, 22,  40,  46,  34,  16,   4,  1;
%e A059259   1,  8, 29,  62,  86,  80,  50,  20,  5,  0;
%e A059259   1,  9, 37,  91, 148, 166, 130,  70, 25,  5, 1;
%e A059259   1, 10, 46, 128, 239, 314, 296, 200, 95, 30, 6, 0;
%e A059259 ...
%p A059259 read transforms; 1/(1-x-x*y-y^2); SERIES2(%,x,y,12); SERIES2TOLIST(%,x,y,12);
%t A059259 T[n_, 0]:= 1; T[n_, n_]:= (1+(-1)^n)/2; T[n_, k_]:= T[n, k] = T[n-1, k] + T[n-1, k-1]; Table[T[n, k], {n, 0, 10} , {k, 0, n}]//Flatten (* _G. C. Greubel_, Jan 03 2017 *)
%o A059259 (Sage)
%o A059259 def A059259_row(n):
%o A059259     @cached_function
%o A059259     def prec(n, k):
%o A059259         if k==n: return (-1)^n
%o A059259         if k==0: return 0
%o A059259         return prec(n-1,k-1)-sum(prec(n,k+i-1) for i in (2..n-k+1))
%o A059259     return [(-1)^(n-k+1)*prec(n+1, k) for k in (1..n)]
%o A059259 for n in (1..12): print(A059259_row(n)) # _Peter Luschny_, Mar 16 2016
%o A059259 (PARI) {T(n,k) = if(k==0, 1, if(k==n, (1+(-1)^n)/2, T(n-1,k) +T(n-1,k-1)) )};
%o A059259 for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Apr 29 2019
%Y A059259 See A059260 for an explicit formula.
%Y A059259 Diagonals of this triangle are given by A006498.
%Y A059259 Similar to the triangles A035317, A080242, A108561, A112555.
%Y A059259 Cf. A123521, A157897.
%K A059259 nonn,tabl
%O A059259 0,8
%A A059259 _N. J. A. Sloane_, Jan 23 2001