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A156586 A new q-combination type general triangle sequence based on Stirling first polynomials: here q=4: m=3: 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])].

%I A156586 #2 Mar 30 2012 17:34:33
%S A156586 1,1,1,1,4,1,1,20,20,1,1,120,600,120,1,1,840,25200,25200,840,1,1,6720,
%T A156586 1411200,8467200,1411200,6720,1,1,60480,101606400,4267468800,
%U A156586 4267468800,101606400,60480,1,1,604800,9144576000,3072577536000,21508042752000
%N A156586 A new q-combination type general triangle sequence based on Stirling first polynomials: here q=4: m=3: 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])].
%C A156586 Row sums are:
%C A156586 {1, 2, 6, 42, 842, 52082, 11303042, 8738271362, 27671488185602,
%C A156586 346773112532985602, 20244862147392528307202,...}.
%C A156586 The q=2 sequence is A009963.
%F A156586 q=4: m=3:
%F A156586 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}]];
%F A156586 b(n,k,m)=If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])].
%e A156586 {1},
%e A156586 {1, 1},
%e A156586 {1, 4, 1},
%e A156586 {1, 20, 20, 1},
%e A156586 {1, 120, 600, 120, 1},
%e A156586 {1, 840, 25200, 25200, 840, 1},
%e A156586 {1, 6720, 1411200, 8467200, 1411200, 6720, 1},
%e A156586 {1, 60480, 101606400, 4267468800, 4267468800, 101606400, 60480, 1},
%e A156586 {1, 604800, 9144576000, 3072577536000, 21508042752000, 3072577536000, 9144576000, 604800, 1},
%e A156586 {1, 6652800, 1005903360000, 3041851760640000, 170343698595840000, 170343698595840000, 3041851760640000, 1005903360000, 6652800, 1},
%e A156586 {1, 79833600, 132779243520000, 4015244324044800000, 2023683139318579200000, 16189465114548633600000, 2023683139318579200000, 4015244324044800000, 132779243520000, 79833600, 1}
%t A156586 Clear[t, n, m, i, k, a, b];
%t A156586 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}]];
%t A156586 b[n_, k_, m_] = If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])];
%t A156586 Table[Flatten[Table[Table[b[n, k, m], {k, 0, n}], {n, 0, 10}]], {m, 0, 15}]
%Y A156586 A009963
%K A156586 nonn,tabl,uned
%O A156586 0,5
%A A156586 _Roger L. Bagula_, Feb 10 2009