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 A156594 A q-Stirling 2nd triangle sequence:q=3;m=2; t(n,k)=If[m == 0, n!, Product[Sum[(-1)^i*StirlingS2[ 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])]. 0
 1, 1, 1, 1, -3, 1, 1, 6, 6, 1, 1, -3, 6, -3, 1, 1, -21, -21, -21, -21, 1, 1, 24, 168, -84, 168, 24, 1, 1, 195, -1560, 5460, 5460, -1560, 195, 1, 1, -111, 7215, 28860, 202020, 28860, 7215, -111, 1, 1, -3072, -113664, -3694080, 29552640, 29552640, -3694080 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Row sums are: {1, 2, -1, 14, 2, -82, 302, 8192, 273950, 51483650, -4132493224,...}. On the sequence only q=2 and q=3 are Integers, the rest have a few rational terms. LINKS FORMULA q=3;m=2; t(n,k)=If[m == 0, n!, Product[Sum[(-1)^i*StirlingS2[ 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])]. EXAMPLE {1}, {1, 1}, {1, -3, 1}, {1, 6, 6, 1}, {1, -3, 6, -3, 1}, {1, -21, -21, -21, -21, 1}, {1, 24, 168, -84, 168, 24, 1}, {1, 195, -1560, 5460, 5460, -1560, 195, 1}, {1, -111, 7215, 28860, 202020, 28860, 7215, -111, 1}, {1, -3072, -113664, -3694080, 29552640, 29552640, -3694080, -113664, -3072, 1}, {1, -4053, -4150272, 76780032, -4990702080, 5703659520, -4990702080, 76780032, -4150272, -4053, 1} MATHEMATICA t[n_, m_] = If[m == 0, n!, Product[Sum[(-1)^i* StirlingS2[k - 1, i]*(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]]; b[n_, k_, m_] = f[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}] CROSSREFS Sequence in context: A159572 A190907 A035582 * A109647 A176668 A054120 Adjacent sequences:  A156591 A156592 A156593 * A156595 A156596 A156597 KEYWORD sign,tabl,uned AUTHOR Roger L. Bagula, Feb 10 2009 STATUS approved

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Last modified April 21 19:16 EDT 2021. Contains 343156 sequences. (Running on oeis4.)