_Roger L. Bagula (rlbagulatftn(AT)yahoo.com), _, Nov 12 2009

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

0, 3, 3, 6, 9, 54, 54, 27, 324, 810, 540, 27, 432, 2322, 3780, 1890, 81, 810, 12150, 42120, 51030, 20412, 243, 3402, 27216, 272160, 697410, 673596, 224532, 243, 34020, 40824, 244944, 1786050, 3633336, 2918916, 833976, 729, 104976, 1583388, 1224720

0,2

Row sums are:

{0, 3, 9, 117, 1701, 8451, 126603, 1898559, 9492309, 142383177, 2135743281...}

m=4;

A(n,k)= (m*n - m*k + 1)A(n - 1, k - 1} + (m*k - (m - 1))A(n - 1, k)

q(n,k)=InverseZTransform[x*Sum[a[[n, k]]*x^(k - 1), {k, 1, n}]/(x - 1)^n, x, k]

out_n,k=3^Floor[(2*k - 1)/3]*coefficients(q[n,k])

{0},

{3},

{3, 6},

{9, 54, 54},

{27, 324, 810, 540},

{27, 432, 2322, 3780, 1890},

{81, 810, 12150, 42120, 51030, 20412},

{243, 3402, 27216, 272160, 697410, 673596, 224532},

{243, 34020, 40824, 244944, 1786050, 3633336, 2918916, 833976},

{729, 104976, 1583388, 1224720, 5664330, 32332608, 54561276, 37528920, 9382230},

{2187, -5734314, 6009876, 53905176, 31689630, 117756828, 551675124, 795613104, 478493730, 106331940}

m = 4; A[n_, 1] := 1; A[n_, n_] := 1

A[n_, k_] := (m*n - m*k + 1)A[n - 1, k - 1] + (m*k - (m - 1))A[n - 1, k]

a = Table[A[n, k], {n, 10}, {k, n}]

p[x_, n_] = x*Sum[a[[n, k]]*x^(k - 1), {k, 1, n}]/(x - 1)

b = Table[p[x, n], {n, 0, 10}]

Table[3^Floor[(2*k - 1)/3]*CoefficientList[ExpandAll[ InverseZTransform[b[[k]], x, n] /. UnitStep[ -1 + n] -> 1], n], {k, 1, Length[b]}]

Cf. A142458, A060187, A008292, A142459

nonn,uned

Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 12 2009

approved