A122950 - OEIS

%I #26 Jan 21 2020 00:40:21

%S 1,0,1,0,0,2,0,0,1,3,0,0,0,3,5,0,0,0,1,7,8,0,0,0,0,4,15,13,0,0,0,0,1,

%T 12,30,21,0,0,0,0,0,5,31,58,34,0,0,0,0,0,1,18,73,109,55,0,0,0,0,0,0,6,

%U 54,162,201,89,0,0,0,0,0,0,1,25,145,344,365,144,0,0,0,0,0,0,0,7,85,361,707

%N Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 0, 1, -1, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, -1, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

%C Skew triangle associated with the Fibonacci numbers.

%H H. Fuks and J. M. G. Soto, <a href="http://arxiv.org/abs/1306.1189">Exponential convergence to equilibrium in cellular automata asymptotically emulating identity</a>, arXiv preprint arXiv:1306.1189 [nlin.CG], 2013.

%F Sum_{k=0..n} T(n,k) = A011782(n).

%F Sum_{n>=k} T(n,k) = A001333(k).

%F T(n,k) = 0 if k < 0 or if k > n, T(0,0) = 1, T(2,1) = 0, T(n,k) = T(n-1,k-1) + T(n-2,k-1) + T(n-2,k-2).

%F T(n,n) = Fibonacci(n+1) = A000045(n+1).

%F Sum_{k=0..n} T(n,k)*x^k = A000007(n), A011782(n), A133592(n), A133594(n), A133642(n), A133646(n), A133678(n), A133679(n), A133680(n), A133681(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively. - _Philippe Deléham_, Jan 03 2008

%F G.f.: (1-y*x^2)/(1-y*x-y*(y+1)*x^2). - _Philippe Deléham_, Nov 26 2011

%e Triangle begins:

%e   1;

%e   0, 1;

%e   0, 0, 2;

%e   0, 0, 1, 3;

%e   0, 0, 0, 3, 5;

%e   0, 0, 0, 1, 7, 8;

%e   0, 0, 0, 0, 4, 15, 13;

%e   0, 0, 0, 0, 1, 12, 30, 21;

%e   0, 0, 0, 0, 0,  5, 31, 58,  34;

%e   0, 0, 0, 0, 0,  1, 18, 73, 109,  55;

%e   0, 0, 0, 0, 0,  0,  6, 54, 162, 201,  89;

%e   0, 0, 0, 0, 0,  0,  1, 25, 145, 344, 365, 144;

%e   0, 0, 0, 0, 0,  0,  0,  7,  85, 361, 707, 655, 233;

%t T[0, 0] = T[1, 1] = 1; T[_, 0] = T[_, 1] = 0; T[n_, n_] := Fibonacci[n+1]; T[n_, k_] /; 0 <= k <= n := T[n, k] = T[n-1, k-1] + T[n-2, k-1] + T[n-2, k-2]; T[_, _] = 0;

%t Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Nov 29 2018 *)

%Y Cf. A055830 (another version).

%K nonn,tabl

%O 0,6

%A _Philippe Deléham_, Oct 25 2006