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A156587 A new q-combination type general triangle sequence based on Stirling first polynomials: here q=5: m=4: 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 A156587 #2 Mar 30 2012 17:34:33
%S A156587 1,1,1,1,5,1,1,30,30,1,1,210,1260,210,1,1,1680,70560,70560,1680,1,1,
%T A156587 15120,5080320,35562240,5080320,15120,1,1,151200,457228800,
%U A156587 25604812800,25604812800,457228800,151200,1,1,1663200,50295168000,25348764672000
%N A156587 A new q-combination type general triangle sequence based on Stirling first polynomials: here q=5: m=4: 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 A156587 Row sums are:
%C A156587 {1, 2, 7, 62, 1682, 144482, 45753122, 52124385602, 253588240382402,
%C A156587 4885227205552108802, 454865349223042267910402,...}.
%C A156587 The q=2 sequence is A009963.
%F A156587 q=5: m=4:
%F A156587 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 A156587 b(n,k,m)=If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])].
%e A156587 {1},
%e A156587 {1, 1},
%e A156587 {1, 5, 1},
%e A156587 {1, 30, 30, 1},
%e A156587 {1, 210, 1260, 210, 1},
%e A156587 {1, 1680, 70560, 70560, 1680, 1},
%e A156587 {1, 15120, 5080320, 35562240, 5080320, 15120, 1},
%e A156587 {1, 151200, 457228800, 25604812800, 25604812800, 457228800, 151200, 1},
%e A156587 {1, 1663200, 50295168000, 25348764672000, 202790117376000, 25348764672000, 50295168000, 1663200, 1},
%e A156587 {1, 19958400, 6638962176000, 33460369367040000, 2409146594426880000, 2409146594426880000, 33460369367040000, 6638962176000, 19958400, 1},
%e A156587 {1, 259459200, 1035678099456000, 57417993833840640000, 41340955560365260800000, 372068600043287347200000, 41340955560365260800000, 57417993833840640000, 1035678099456000, 259459200, 1}
%t A156587 Clear[t, n, m, i, k, a, b];
%t A156587 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 A156587 b[n_, k_, m_] = If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])];
%t A156587 Table[Flatten[Table[Table[b[n, k, m], {k, 0, n}], {n, 0, 10}]], {m, 0, 15}]
%Y A156587 A009963
%K A156587 nonn,tabl,uned
%O A156587 0,5
%A A156587 _Roger L. Bagula_, Feb 10 2009