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A094416
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Array read by antidiagonals: generalized ordered Bell numbers Bo(r,n).
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19
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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, 7087261
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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-r*x).
Also, Bo(r,s) is ((x*d/dx)^n)(1/(1+r-r*x)) 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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FORMULA
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E.g.f.: 1/(1 + r*(1 - exp(x))).
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.
Recurrence: Bo(r, n) = r * Sum_{k=1..n} C(n, k)*Bo(r, n-k), with Bo(r, 0) = 1.
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
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EXAMPLE
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Array begins as:
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}];
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PROG
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(Magma)
A094416:= func< n, k | (&+[Factorial(j)*n^j*StirlingSecond(k, j): j in [0..k]]) >;
(SageMath)
def A094416(n, k): return sum(factorial(j)*n^j*stirling_number2(k, j) for j in range(k+1)) # array
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
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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