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A094416 Array read by antidiagonals: generalized ordered Bell numbers Bo(r,n). 17
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; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

LINKS

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

FORMULA

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.

EXAMPLE

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, ...

MATHEMATICA

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 *)

CROSSREFS

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

KEYWORD

nonn,tabl

AUTHOR

Ralf Stephan, May 02 2004

EXTENSIONS

Offset corrected by Geoffrey Critzer, Jan 01 2019

STATUS

approved

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Last modified April 19 07:36 EDT 2021. Contains 343109 sequences. (Running on oeis4.)