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A094416 Array read by antidiagonals: generalized ordered Bell numbers Bo(r,n).

%I A094416 #55 May 12 2024 18:20:41
%S A094416 1,2,3,3,10,13,4,21,74,75,5,36,219,730,541,6,55,484,3045,9002,4683,7,
%T A094416 78,905,8676,52923,133210,47293,8,105,1518,19855,194404,1103781,
%U A094416 2299754,545835,9,136,2359,39390,544505,5227236,26857659,45375130,7087261
%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-r*x).
%C A094416 Also, Bo(r,s) is ((x*d/dx)^n)(1/(1+r-r*x)) evaluated at x=1.
%C A094416 r-th ordered Bell polynomial (A019538) evaluated at n.
%C A094416 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
%C A094416 Row r (starting at r=0), Bo(r+1, n), is the Akiyama-Tanigawa algorithm applied to the powers of r+1. See Python program below. - _Shel Kaphan_, May 03 2024
%H A094416 G. C. Greubel, <a href="/A094416/b094416.txt">Antidiagonals n = 1..50, flattened</a>
%H A094416 Paul Barry, <a href="https://arxiv.org/abs/1803.06408">Three Études on a sequence transformation pipeline</a>, arXiv:1803.06408 [math.CO], 2018.
%H A094416 P. Blasiak, K. A. Penson and A. I. Solomon, <a href="https://arxiv.org/abs/quant-ph/0303030">Dobinski-type relations and the log-normal distribution</a>, arXiv:quant-ph/0303030, 2003.
%H A094416 C. G. Bower, <a href="/transforms2.html">Transforms</a>
%F A094416 E.g.f.: 1/(1 + r*(1 - exp(x))).
%F A094416 Bo(r, n) = Sum_{k=0..n} k!*r^k*Stirling2(n, k) = 1/(r+1) * Sum_{k>=1} k^n * (r/(r+1))^k, for r>0, n>0.
%F A094416 Recurrence: Bo(r, n) = r * Sum_{k=1..n} C(n, k)*Bo(r, n-k), with Bo(r, 0) = 1.
%F A094416 Bo(r,0) = 1, Bo(r,n) = r*Bo(r,n-1) - (r+1)*Sum_{j=1..n-1} (-1)^j * binomial(n-1,j) * Bo(r,n-j). - _Seiichi Manyama_, Nov 17 2023
%e A094416 Array begins as:
%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, ...
%t A094416 Bo[_, 0]=1; Bo[r_, n_]:= Bo[r, n]= r*Sum[Binomial[n,k] Bo[r,n-k], {k, n}];
%t A094416 Table[Bo[r-n+1, n], {r, 10}, {n, r}] // Flatten (* _Jean-François Alcover_, Nov 03 2018 *)
%o A094416 (Magma)
%o A094416 A094416:= func< n,k | (&+[Factorial(j)*n^j*StirlingSecond(k,j): j in [0..k]]) >;
%o A094416 [A094416(n-k+1,k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Jan 12 2024
%o A094416 (SageMath)
%o A094416 def A094416(n,k): return sum(factorial(j)*n^j*stirling_number2(k,j) for j in range(k+1)) # array
%o A094416 flatten([[A094416(n-k+1,k) for k in range(1,n+1)] for n in range(1,13)]) # _G. C. Greubel_, Jan 12 2024
%o A094416 (Python)
%o A094416 # The Akiyama-Tanigawa algorithm applied to the powers of r + 1
%o A094416 # generates the rows. Adds one row (r=0) and one column (n=0).
%o A094416 # Adapted from Peter Luschny on A371568.
%o A094416 def f(n, r): return (r + 1)**n
%o A094416 def ATtransform(r, len, f):
%o A094416   A = [0] * len
%o A094416   R = [0] * len
%o A094416   for n in range(len):
%o A094416       R[n] = f(n, r)
%o A094416       for j in range(n, 0, -1):
%o A094416           R[j - 1] = j * (R[j] - R[j - 1])
%o A094416       A[n] = R[0]
%o A094416   return A
%o A094416 for r in range(8): print([r], ATtransform(r, 8, f)) # _Shel Kaphan_, May 03 2024
%Y A094416 Rows 1-10 are A000670, A004123, A032033, A094417, A094418, A094419, A238464, A238465, A238466, A238467.
%Y A094416 Columns include A014105, A094421.
%Y A094416 Main diagonal is A094420.
%Y A094416 Antidiagonal sums are A094422.
%Y A094416 Cf. A019538, A131689, A344499.
%K A094416 nonn,tabl
%O A094416 1,2
%A A094416 _Ralf Stephan_, May 02 2004
%E A094416 Offset corrected by _Geoffrey Critzer_, Jan 01 2019