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A167787 Triangle of z Transform coefficients from General Pascal [1,10,1} A142459 polynomials multiplied by factor 3^Floor[(2*k - 1)/3].

%I A167787 #2 Mar 30 2012 17:34:35
%S A167787 0,3,3,6,9,54,54,27,324,810,540,27,432,2322,3780,1890,81,810,12150,
%T A167787 42120,51030,20412,243,3402,27216,272160,697410,673596,224532,243,
%U A167787 34020,40824,244944,1786050,3633336,2918916,833976,729,104976,1583388,1224720
%N A167787 Triangle of z Transform coefficients from General Pascal [1,10,1} A142459 polynomials multiplied by factor 3^Floor[(2*k - 1)/3].
%C A167787 Row sums are:
%C A167787 {0, 3, 9, 117, 1701, 8451, 126603, 1898559, 9492309, 142383177, 2135743281...}
%F A167787 m=4;
%F A167787 A(n,k)= (m*n - m*k + 1)A(n - 1, k - 1} + (m*k - (m - 1))A(n - 1, k)
%F A167787 q(n,k)=InverseZTransform[x*Sum[a[[n, k]]*x^(k - 1), {k, 1, n}]/(x - 1)^n, x, k]
%F A167787 out_n,k=3^Floor[(2*k - 1)/3]*coefficients(q[n,k])
%e A167787 {0},
%e A167787 {3},
%e A167787 {3, 6},
%e A167787 {9, 54, 54},
%e A167787 {27, 324, 810, 540},
%e A167787 {27, 432, 2322, 3780, 1890},
%e A167787 {81, 810, 12150, 42120, 51030, 20412},
%e A167787 {243, 3402, 27216, 272160, 697410, 673596, 224532},
%e A167787 {243, 34020, 40824, 244944, 1786050, 3633336, 2918916, 833976},
%e A167787 {729, 104976, 1583388, 1224720, 5664330, 32332608, 54561276, 37528920, 9382230},
%e A167787 {2187, -5734314, 6009876, 53905176, 31689630, 117756828, 551675124, 795613104, 478493730, 106331940}
%t A167787 m = 4; A[n_, 1] := 1; A[n_, n_] := 1
%t A167787 A[n_, k_] := (m*n - m*k + 1)A[n - 1, k - 1] + (m*k - (m - 1))A[n - 1, k]
%t A167787 a = Table[A[n, k], {n, 10}, {k, n}]
%t A167787 p[x_, n_] = x*Sum[a[[n, k]]*x^(k - 1), {k, 1, n}]/(x - 1)
%t A167787 b = Table[p[x, n], {n, 0, 10}]
%t A167787 Table[3^Floor[(2*k - 1)/3]*CoefficientList[ExpandAll[ InverseZTransform[b[[k]], x, n] /. UnitStep[ -1 + n] -> 1], n], {k, 1, Length[b]}]
%Y A167787 Cf. A142458, A060187, A008292, A142459
%K A167787 nonn,uned
%O A167787 0,2
%A A167787 _Roger L. Bagula_, Nov 12 2009