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Search: id:A094416
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| A094416 |
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Array read by antidiagonals: generalized ordered Bell numbers Bo(r,n). |
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12
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| 1, 2, 3, 3, 10, 13, 4, 21, 74, 75, 5, 36, 219, 730, 541, 6, 55, 484, 3045, 9002, 4683, 7, 78, 905, 8676, 52923, 133210, 47293, 8, 105, 1518, 19855, 194404, 1103781, 2299754, 545835, 9, 136, 2359, 39390, 544505, 5227236, 26857659, 45375130
(list; table; graph; listen)
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OFFSET
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0,2
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COMMENT
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Also, r times the number of (r+1)-level labeled linear rooted trees with n leaves.
"AIJ" (ordered, indistinct, labeled) transform of {r,r,r,...}.
Stirling transform of r^n*n!, i.e. of e.g.f. 1/(1-rx).
Also, Bo(r,s) is ((x*d/dx)^n)(1/(r+1-rx)) evaluated at x=1.
r-th ordered Bell polynomial (A019538) evaluated at n.
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LINKS
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P. Blasiak, K. A. Penson and A. I. Solomon, Dobinski-type relations and the log-normal distribution.
C. G. Bower, Transforms
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FORMULA
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E.g.f.: 1/(r+1-re^x).
Bo(r, n) = Sum[k=0..n, k!*r^k*Stirling2(n, k)] = 1/(r+1)*Sum[k=1..inf, k^n*{r/(r+1)}^k], r>0, n>0.
Recurrence: Bo(r, n) = r*Sum[k=1..n, C(n, k)*Bo(r, n-k)], Bo(r, 0)=1.
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EXAMPLE
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1,3,13,75,541,4683,47293,
2,10,74,730,9002,133210,2299754,
3,21,219,3045,52923,1103781,26857659,
4,36,484,8676,194404,5227236,163978084,
5,55,905,19855,544505,17919055,687978905,
6,78,1518,39390,1277646,49729758,2258233998,
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CROSSREFS
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Rows 1-6 are A000670, A004123, A032033, A094417, A094418, A094419. Columns include A014105, A094421. Main diagonal is A094420. Antidiagonal sums are A094422.
Sequence in context: A110042 A123027 A100652 this_sequence A117030 A009097 A112858
Adjacent sequences: A094413 A094414 A094415 this_sequence A094417 A094418 A094419
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KEYWORD
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nonn,tabl
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AUTHOR
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Ralf Stephan, May 02 2004
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