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

0,18

Comments

Motivation: The exponential transform applied n times to the constant function 1 evaluated at k was studied by E. T. Bell (Iterated Bell numbers, see A144150).

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

Table of n, a(n) for n=0..66.

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.

Example

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

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);

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 *)

Crossrefs

Cf. A000587, A144150, A209631.

Sequence in context: A025902 A219923 A286950 * A272903 A321458 A226194

Adjacent sequences: A210635 A210636 A210637 * A210639 A210640 A210641

Keyword

sign,tabl

Author

Peter Luschny, Mar 26 2012

Status

approved