A210638 Iterated Rényi numbers. Square array A(n,k), n>=0, k>=0, read by antidiagonals, A(n,k) = exponential transform applied n times to the constant function -1, evaluated at k.
-1, 1, -1, 1, -1, -1, 1, -1, 0, -1, 1, -1, 1, 1, -1, 1, -1, 2, 0, 1, -1, 1, -1, 3, -4, -2, -2, -1, 1, -1, 4, -11, 8, 2, -9, -1, 1, -1, 5, -21, 49, -14, 9, -9, -1, 1, -1, 6, -34, 139, -255, 13, -24, 50, -1, 1, -1, 7, -50, 296, -1106, 1508, 45, -80, 267, -1, 1
Offset: 0
Examples
n\k [0] [1] [2] [3] [4] [5] [6] [0] -1 -1 -1 -1 -1 -1 -1 [1] 1 -1 0 1 1 -2 -9 [A000587] [2] 1 -1 1 0 -2 2 9 [3] 1 -1 2 -4 8 -14 13 [4] 1 -1 3 -11 49 -255 1508 [5] 1 -1 4 -21 139 -1106 10244 [6] 1 -1 5 -34 296 -3132 38916 column3(n) = (-2+7*n-3*n^2)/2 [A115067] column4(n) = (-2+21*n-23*n^2+6*n^3)/2 column5(n) = (-6+199*n-405*n^2+245*n^3-45*n^4)/6 column6(n) = (-24+2866*n-9213*n^2+9470*n^3-3855*n^4+540*n^5 )/24
References
- R. E. Beard, On the coefficients in the expansion of e^e^t and e^e^(-t), J. Inst. Actuar. 76 (1950), 152-163.
- Alfréd Rényi, New methods and results in combinatorial analysis. (Paper is in Hungarian.) I. MTA III Oszt. Ivozl., 16 (1966), 77-105.
Links
- E. T. Bell, The iterated exponential integers, Ann. Math. 39(3) (1938), 539-557.
- Antal E. Fekete, Apropos Bell and Stirling Numbers, Crux Mathematicorum with Mathematical Mayhem, Canadian Mathematical Society, Volume 25 Number 5 (May 1999), 274-281.
- Peter Luschny, Set partitions and Bell numbers
- V. R. Rao Uppuluri and J. A. Carpenter, Numbers generated by the function exp(1-e^x), Fib. Quart. 7 (1969), 437-448.
Programs
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Maple
exptr := proc(p) local g; g := proc(n) option remember; local k; `if`(n=0, 1, add(binomial(n-1, k-1)*p(k)*g(n-k), k=1..n)) end end: A210638 := (n, k) -> (exptr@@n)(-1)(k): seq(lprint(seq(A210638(n, k), k=0..6)), n=0..6);
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Mathematica
exptr[p_] := Module[{g}, g[n_] := g[n] = If[n==0, 1, Sum[Binomial[n-1, k-1] p[k] g[n-k], {k, 1, n}]]; g]; A[n_, k_] := Nest[exptr, -1&, n][k]; Table[A[n-k, k], {n, 0, 10}, {k, 0, n}] (* Jean-François Alcover, Jun 29 2019 *)
Comments