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A207629 Triangle of coefficients of polynomials u(n,x) jointly generated with A207630; see the Formula section.

%I A207629 #14 Dec 25 2022 13:59:18
%S A207629 1,2,5,1,11,4,23,13,1,47,37,6,95,97,25,1,191,241,87,8,383,577,271,41,
%T A207629 1,767,1345,783,169,10,1535,3073,2143,609,61,1,3071,6913,5631,2001,
%U A207629 291,12,6143,15361,14335,6145,1191,85,1,12287,33793,35583,17921
%N A207629 Triangle of coefficients of polynomials u(n,x) jointly generated with A207630; see the Formula section.
%C A207629 With offset 0, equals the stretched Riordan array ((1 - z + z^2)/(1 - 3*z + 2*z^2), z^2/(1 - 2*z)) in the notation of Corsani et al., Section 2. - _Peter Bala_, Dec 31 2015
%H A207629 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 A207629 u(n,x) = u(n-1,x) + v(n-1,x),
%F A207629 v(n,x) = (x + 1)*u(n-1,x) + v(n-1,x) + 1,
%F A207629 where u(1,x) = 1, v(1,x) = 1.
%e A207629 First five rows:
%e A207629    1
%e A207629    2
%e A207629    5  1
%e A207629   11  4
%e A207629   23 13  1
%t A207629 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A207629 u[n_, x_] := u[n - 1, x] + v[n - 1, x]
%t A207629 v[n_, x_] := (x + 1)*u[n - 1, x] + v[n - 1, x] + 1
%t A207629 Table[Factor[u[n, x]], {n, 1, z}]
%t A207629 Table[Factor[v[n, x]], {n, 1, z}]
%t A207629 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A207629 TableForm[cu]
%t A207629 Flatten[%]    (* A207629 *)
%t A207629 Table[Expand[v[n, x]], {n, 1, z}]
%t A207629 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A207629 TableForm[cv]
%t A207629 Flatten[%]    (* A207630 *)
%Y A207629 Cf. A207630, A208510, A083329 (column 1).
%K A207629 nonn,tabf,easy
%O A207629 1,2
%A A207629 _Clark Kimberling_, Feb 23 2012