1,5

Row sums are:{1, 2, 7, 30, 157, 972, 6961, 56660, 516901, 5225670}.

I put the new quantum Pascal (two term)

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

together with the (three term) used by Paul Barry

A(n.k)=A(n - 1, k - 1) + A(n - 1, k) + j*A(n - 2, k - 1)

to get the new recursion:

A(n.k)=(m*n - m*k + 1)A(n - 1, k - 1) + (m*k - (m - 1))A(n - 1, k) + j*A(n - 2, k - 1).

The result seems to be completely new and almost unexplored.

m=j=1; A(n.k)=(m*n - m*k + 1)A(n - 1, k - 1) + (m*k - (m - 1))A(n - 1, k) + j*A(n - 2, k - 1).

{1},

{1, 1},

{1, 5, 1},

{1, 14, 14, 1},

{1, 33, 89, 33, 1},

{1, 72, 413, 413, 72, 1},

{1, 151, 1632, 3393, 1632, 151, 1},

{1, 310, 5874, 22145, 22145, 5874, 310, 1},

{1, 629, 19943, 125456, 224843, 125456, 19943, 629, 1},

{1, 1268, 65171, 647299, 1899096, 1899096, 647299, 65171, 1268, 1}

Clear[A, m, n, j, k] m = 1 = j = 1; 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] + j*A[n - 2, k - 1]; a = Table[A[n, k], {n, 10}, {k, n}]; Flatten[a]

Sequence in context: A157177 A119725 A111910 this_sequence A157207 A008957 A136267

Adjacent sequences: A144435 A144436 A144437 this_sequence A144439 A144440 A144441

nonn,uned

Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Oct 05 2008