A207616 - OEIS

%I #17 Dec 25 2022 13:59:01

%S 1,2,4,1,7,4,11,11,1,16,25,6,22,50,22,1,29,91,63,8,37,154,154,37,1,46,

%T 246,336,129,10,56,375,672,375,56,1,67,550,1254,957,231,12,79,781,

%U 2211,2211,781,79,1,92,1079,3718,4719,2288,377,14,106,1456,6006

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A207617; see the Formula section.

%C With offset 0, equals the stretched Riordan array ((1 - z + z^2)/(1 - z)^3, z^2/(1 - z)^2) in the notation of Corsani et al., Section 2. - _Peter Bala_, Dec 31 2015

%H C. Corsani, D. Merlini, and R. Sprugnoli, <a href="http://dx.doi.org/10.1016/S0012-365X(97)00110-6">Left-inversion of combinatorial sums</a>, Discrete Mathematics, 180 (1998) 107-122.

%F u(n,x) = u(n-1,x) + v(n-1,x), v(n,x) = x*u(n-1,x) + v(n-1,x) + 1, where u(1,x) = 1, v(1,x) = 1.

%e First five rows:

%e    1

%e    2

%e    4  1

%e    7  4

%e   11 11  1

%t u[1, x_] := 1; v[1, x_] := 1; z = 16;

%t u[n_, x_] := u[n - 1, x] + v[n - 1, x]

%t v[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1

%t Table[Factor[u[n, x]], {n, 1, z}]

%t Table[Factor[v[n, x]], {n, 1, z}]

%t cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];

%t TableForm[cu]

%t Flatten[%]    (* A207616 *)

%t Table[Expand[v[n, x]], {n, 1, z}]

%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

%t TableForm[cv]

%t Flatten[%]    (* A207617 *)

%Y Cf. A207617, A208510, A000124 (column 1).

%K nonn,tabf,easy

%O 1,2

%A _Clark Kimberling_, Feb 20 2012