Search: id:A094416
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%I A094416
%S A094416 1,2,3,3,10,13,4,21,74,75,5,36,219,730,541,6,55,484,3045,9002,4683,7,78,
%T A094416 905,8676,52923,133210,47293,8,105,1518,19855,194404,1103781,2299754,
%U A094416 545835,9,136,2359,39390,544505,5227236,26857659,45375130
%N A094416 Array read by antidiagonals: generalized ordered Bell numbers Bo(r,n).
%C A094416 Also, r times the number of (r+1)-level labeled linear rooted trees with 
               n leaves.
%C A094416 "AIJ" (ordered, indistinct, labeled) transform of {r,r,r,...}.
%C A094416 Stirling transform of r^n*n!, i.e. of e.g.f. 1/(1-rx).
%C A094416 Also, Bo(r,s) is ((x*d/dx)^n)(1/(r+1-rx)) evaluated at x=1.
%C A094416 r-th ordered Bell polynomial (A019538) evaluated at n.
%H A094416 P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://arXiv.org/
               abs/quant-ph/0303030">Dobinski-type relations and the log-normal 
               distribution</a>.
%H A094416 C. G. Bower, <a href="transforms2.html">Transforms</a>
%F A094416 E.g.f.: 1/(r+1-re^x).
%F A094416 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.
%F A094416 Recurrence: Bo(r, n) = r*Sum[k=1..n, C(n, k)*Bo(r, n-k)], Bo(r, 0)=1.
%e A094416 1,3,13,75,541,4683,47293,
%e A094416 2,10,74,730,9002,133210,2299754,
%e A094416 3,21,219,3045,52923,1103781,26857659,
%e A094416 4,36,484,8676,194404,5227236,163978084,
%e A094416 5,55,905,19855,544505,17919055,687978905,
%e A094416 6,78,1518,39390,1277646,49729758,2258233998,
%Y A094416 Rows 1-6 are A000670, A004123, A032033, A094417, A094418, A094419. Columns 
               include A014105, A094421. Main diagonal is A094420. Antidiagonal 
               sums are A094422.
%Y A094416 Cf. A131689 [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 03 2008]
%Y A094416 Sequence in context: A110042 A123027 A100652 this_sequence A152300 A117030 
               A155758
%Y A094416 Adjacent sequences: A094413 A094414 A094415 this_sequence A094417 A094418 
               A094419
%K A094416 nonn,tabl
%O A094416 0,2
%A A094416 Ralf Stephan, May 02 2004

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