0,5

Row sums are:

{1, 2, 5, 26, 362, 15122, 2121842, 1049973122, 2050823940482,

15854719559212802, 552278629803518956802,...}.

The q=2 sequence is A009963.

I had to adjust the sign to get an all positive set of sequences.

I don't get any of the others in OEIS yet.

q=3: m=2:

t(n,k)=If[m == 0, n!, Product[Sum[(-1)^(i + k)*StirlingS1[k - 1, i]*(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]];

b(n,k,m)=If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])].

{1},

{1, 1},

{1, 3, 1},

{1, 12, 12, 1},

{1, 60, 240, 60, 1},

{1, 360, 7200, 7200, 360, 1},

{1, 2520, 302400, 1512000, 302400, 2520, 1},

{1, 20160, 16934400, 508032000, 508032000, 16934400, 20160, 1},

{1, 181440, 1219276800, 256048128000, 1536288768000, 256048128000, 1219276800, 181440, 1},

{1, 1814400, 109734912000, 184354652160000, 7742895390720000, 7742895390720000, 184354652160000, 109734912000, 1814400, 1},

{1, 19958400, 12070840320000, 182511105638400000, 61323731494502400000, 429266120461516800000, 61323731494502400000, 182511105638400000, 12070840320000, 19958400, 1}

Clear[t, n, m, i, k, a, b];

t[n_, m_] = If[m == 0, n!, Product[Sum[(-1)^(i + k)*StirlingS1[k - 1, i]*(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]];

b[n_, k_, m_] = If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])];

Table[Flatten[Table[Table[b[n, k, m], {k, 0, n}], {n, 0, 10}]], {m, 0, 15}]

A009963

Sequence in context: A098778 A078122 A128592 this_sequence A129619 A094573 A055154

Adjacent sequences: A156581 A156582 A156583 this_sequence A156585 A156586 A156587

nonn,tabl,uned

Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 10 2009