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A207616 Triangle of coefficients of polynomials u(n,x) jointly generated with A207617; see the Formula section.

%I A207616 #17 Dec 25 2022 13:59:01
%S A207616 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 A207616 246,336,129,10,56,375,672,375,56,1,67,550,1254,957,231,12,79,781,
%U A207616 2211,2211,781,79,1,92,1079,3718,4719,2288,377,14,106,1456,6006
%N A207616 Triangle of coefficients of polynomials u(n,x) jointly generated with A207617; see the Formula section.
%C A207616 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 A207616 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 A207616 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 A207616 First five rows:
%e A207616    1
%e A207616    2
%e A207616    4  1
%e A207616    7  4
%e A207616   11 11  1
%t A207616 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A207616 u[n_, x_] := u[n - 1, x] + v[n - 1, x]
%t A207616 v[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1
%t A207616 Table[Factor[u[n, x]], {n, 1, z}]
%t A207616 Table[Factor[v[n, x]], {n, 1, z}]
%t A207616 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A207616 TableForm[cu]
%t A207616 Flatten[%]    (* A207616 *)
%t A207616 Table[Expand[v[n, x]], {n, 1, z}]
%t A207616 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A207616 TableForm[cv]
%t A207616 Flatten[%]    (* A207617 *)
%Y A207616 Cf. A207617, A208510, A000124 (column 1).
%K A207616 nonn,tabf,easy
%O A207616 1,2
%A A207616 _Clark Kimberling_, Feb 20 2012