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A167786 Triangle of z Transform coefficients from General Pascal [1,8,1} A142458 polynomials multiplied by factor 3^Floor[(2*k - 1)/3].

%I A167786 #2 Mar 30 2012 17:34:35
%S A167786 0,3,3,6,9,45,45,27,234,540,360,27,315,1305,1980,990,81,1026,6750,
%T A167786 18360,20790,8316,243,3807,26379,115830,234630,212058,70686,243,7938,
%U A167786 37800,177660,582120,939708,706860,201960,729,26001,280827,873180,3087315
%N A167786 Triangle of z Transform coefficients from General Pascal [1,8,1} A142458 polynomials multiplied by factor 3^Floor[(2*k - 1)/3].
%C A167786 Row sums are:
%C A167786 {0, 3, 9, 99, 1161, 4617, 55323, 663633, 2654289, 31850739, 382206681...}
%C A167786 These are a sequence of Infinite sums that give A142458.
%C A167786 Even terms are factored by (1+2*n) which is the MacMahon (1+2*n)^k,but the polynomials seem fundamental
%C A167786 other than that.
%C A167786 A060187 MacMahon gives A013609 Triangle of coefficients in expansion of (1 + 2x)^n.
%C A167786 I looked for a simple infinite sum for the {1,8,1} and failed.
%C A167786 This reasoning comes from finding that the general z Transform polynomials are
%C A167786 related to the Eulerian: in fact this type of Eulerian polynomials A008292 gives (1+n)^k binomial.
%C A167786 The polynomials given here form a set of infinite sum sequences.
%F A167786 m=3;
%F A167786 A(n,k)= (m*n - m*k + 1)A(n - 1, k - 1} + (m*k - (m - 1))A(n - 1, k)
%F A167786 q(n,k)=InverseZTransform[x*Sum[a[[n, k]]*x^(k - 1), {k, 1, n}]/(x - 1)^n, x, k]
%F A167786 out_n,k=3^Floor[(2*k - 1)/3]*coefficients(q[n,k])
%e A167786 {0},
%e A167786 {3},
%e A167786 {3, 6},
%e A167786 {9, 45, 45},
%e A167786 {27, 234, 540, 360},
%e A167786 {27, 315, 1305, 1980, 990},
%e A167786 {81, 1026, 6750, 18360, 20790, 8316},
%e A167786 {243, 3807, 26379, 115830, 234630, 212058, 70686},
%e A167786 {243, 7938, 37800, 177660, 582120, 939708, 706860, 201960},
%e A167786 {729, 26001, 280827, 873180, 3087315, 8058204, 10814958, 6967620, 1741905},
%e A167786 {2187, -308610, 1076490, 7334820, 17120565, 48411594, 104968710, 120570120, 67934295, 15096510}
%t A167786 m = 3 A[n_, 1] := 1 A[n_, n_] := 1
%t A167786 A[n_, k_] := (m*n - m*k + 1)A[n - 1, k - 1] + (m*k - (m - 1))A[n - 1, k]
%t A167786 a = Table[A[n, k], {n, 10}, {k, n}]
%t A167786 p[x_, n_] = x*Sum[a[[n, k]]*x^(k - 1), {k, 1, n}]/(x - 1)
%t A167786 b = Table[p[x, n], {n, 0, 10}]
%t A167786 Table[3^Floor[(2*k - 1)/3]*CoefficientList[ExpandAll[ InverseZTransform[b[[k]], x, n] /. UnitStep[ -1 + n] -> 1], n], {k, 1, Length[b]}]
%Y A167786 A142458, A060187, A008292
%K A167786 nonn,uned
%O A167786 0,2
%A A167786 _Roger L. Bagula_, Nov 12 2009