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A094416
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Array read by antidiagonals: generalized ordered Bell numbers Bo(r,n).
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17
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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
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OFFSET
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1,2
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COMMENTS
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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.
Bo(r,n) is the n-th moment of a geometric distribution with probability parameter = 1/(r+1). Here, geometric distribution is the number of failures prior to the first success. - Geoffrey Critzer, Jan 01 2019
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LINKS
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Table of n, a(n) for n=1..44.
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
P. Blasiak, K. A. Penson and A. I. Solomon, Dobinski-type relations and the log-normal distribution, arXiv:quant-ph/0303030, 2003.
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), with r>0, n>0.
Recurrence: Bo(r, n) = r * sum( k=1..n, C(n, k)*Bo(r, n-k) ), with 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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MATHEMATICA
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Bo[_, 0]=1; Bo[r_, n_] := Bo[r, n] = r Sum[Binomial[n, k] Bo[r, n-k], {k, n}];
Table[Bo[r-n+1, n], {r, 10}, {n, r}] // Flatten (* Jean-François Alcover, Nov 03 2018 *)
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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.
Cf. A131689. [Philippe Deléham, Nov 03 2008]
Sequence in context: A337432 A123027 A100652 * A218868 A329874 A152300
Adjacent sequences: A094413 A094414 A094415 * 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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EXTENSIONS
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Offset corrected by Geoffrey Critzer, Jan 01 2019
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STATUS
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approved
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