login
Riordan array (1/(1-x-x^2), x/(1+2*x)).
1

%I #17 Mar 15 2017 04:00:20

%S 1,1,1,2,-1,1,3,4,-3,1,5,-5,10,-5,1,8,15,-25,20,-7,1,13,-22,65,-65,34,

%T -9,1,21,57,-152,195,-133,52,-11,1,34,-93,361,-542,461,-237,74,-13,1,

%U 55,220,-815,1445,-1464,935,-385,100,-15,1,89,-385,1850,-3705

%N Riordan array (1/(1-x-x^2), x/(1+2*x)).

%C First column: Fibonacci numbers A000045(n+1).

%H Indranil Ghosh, <a href="/A237498/b237498.txt">Rows 0..100, flattened</a>

%F Sum_{k=0..n} T(n,k)*x^k = A000045(n+1), A098600(n), A000032(n+1), A027961(n+1), A027974(n) for x = 0, 1, 2, 3, 4 respectively.

%F T(n,k) = T(n-1,k-1) - T(n-1,k) + 3*T(n-2,k) - T(n-2,k-1) + 2*T(n-3,k) - T(n-3,k-1), T(0,0) = T(1,0) = T(1,1) = T(2,2) = 1, T(2,0) = 2, T(2,1) = -1, T(n,k) = 0 if k<0 or if k>n.

%F T(n,0) = T(n-1,0) + T(n-2,0) with T(0,0) = T(1,0) = 1, T(n,k) = T(n-1,k-1) - 2*T(n-1,k) for k>=1.

%F G.f.: (1+2*x)/((1+2*x-y*x)*(1-x-x^2)).

%e Triangle begins:

%e 1;

%e 1, 1;

%e 2, -1, 1;

%e 3, 4, -3, 1;

%e 5, -5, 10, -5, 1;

%e 8, 15, -25, 20, -7, 1;

%e 13, -22, 65, -65, 34, -9, 1;

%e ...

%e Production matrix is:

%e 1, 1;

%e 1, -2, 1;

%e 2, 0, -2, 1;

%e 4, 0, 0, -2, 1;

%e 8, 0, 0, 0, -2, 1;

%e 16, 0, 0, 0, 0, -2, 1;

%e 32, 0, 0, 0, 0, 0, -2, 1;

%e 64, 0, 0, 0, 0, 0, 0, -2, 1;

%e ...

%t nmax=10;Flatten[CoefficientList[Series[CoefficientList[Series[(1 + 2*x) / ((1 + 2*x - y*x) * (1 - x - x^2)), {x, 0, nmax }], x], {y, 0, nmax}], y]] (* _Indranil Ghosh_, Mar 15 2017 *)

%Y Columns: A000045, A084179.

%K easy,sign,tabl

%O 0,4

%A _Philippe Deléham_, Feb 08 2014