cp's OEIS Frontend

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.

%I A210638 #21 Jan 05 2025 19:51:39
%S A210638 -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,
%T A210638 -1,1,-1,4,-11,8,2,-9,-1,1,-1,5,-21,49,-14,9,-9,-1,1,-1,6,-34,139,
%U A210638 -255,13,-24,50,-1,1,-1,7,-50,296,-1106,1508,45,-80,267,-1,1
%N 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.
%C A210638 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).
%D A210638 R. E. Beard, On the coefficients in the expansion of e^e^t and e^e^(-t), J. Inst. Actuar. 76 (1950), 152-163.
%D A210638 Alfréd Rényi, New methods and results in combinatorial analysis. (Paper is in Hungarian.) I. MTA III Oszt. Ivozl., 16 (1966), 77-105.
%H A210638 E. T. Bell, <a href="http://www.jstor.org/stable/1968633">The iterated exponential integers</a>, Ann. Math. 39(3) (1938), 539-557.
%H A210638 Antal E. Fekete, <a href="https://cms.math.ca/crux/v25/n5/page274-281.pdf">Apropos Bell and Stirling Numbers</a>, Crux Mathematicorum with Mathematical Mayhem, Canadian Mathematical Society, Volume 25 Number 5 (May 1999), 274-281.
%H A210638 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/SetPartitions">Set partitions and Bell numbers</a>
%H A210638 V. R. Rao Uppuluri and J. A. Carpenter, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/7-4/uppuluri.pdf">Numbers generated by the function exp(1-e^x)</a>, Fib. Quart. 7 (1969), 437-448.
%e A210638 n\k [0]  [1] [2]   [3]   [4]     [5]     [6]
%e A210638 [0] -1   -1  -1    -1    -1      -1      -1
%e A210638 [1]  1   -1   0     1     1      -2      -9  [A000587]
%e A210638 [2]  1   -1   1     0    -2       2       9
%e A210638 [3]  1   -1   2    -4     8     -14      13
%e A210638 [4]  1   -1   3   -11    49    -255    1508
%e A210638 [5]  1   -1   4   -21   139   -1106   10244
%e A210638 [6]  1   -1   5   -34   296   -3132   38916
%e A210638 column3(n) = (-2+7*n-3*n^2)/2  [A115067]
%e A210638 column4(n) = (-2+21*n-23*n^2+6*n^3)/2
%e A210638 column5(n) = (-6+199*n-405*n^2+245*n^3-45*n^4)/6
%e A210638 column6(n) = (-24+2866*n-9213*n^2+9470*n^3-3855*n^4+540*n^5 )/24
%p A210638 exptr := proc(p) local g; g := proc(n) option remember; local k;
%p A210638 `if`(n=0, 1, add(binomial(n-1, k-1)*p(k)*g(n-k), k=1..n)) end end:
%p A210638 A210638 := (n, k) -> (exptr@@n)(-1)(k):
%p A210638 seq(lprint(seq(A210638(n, k), k=0..6)), n=0..6);
%t A210638 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];
%t A210638 A[n_, k_] := Nest[exptr, -1&, n][k];
%t A210638 Table[A[n-k, k], {n, 0, 10}, {k, 0, n}] (* _Jean-François Alcover_, Jun 29 2019 *)
%Y A210638 Cf. A000587, A144150, A209631.
%K A210638 sign,tabl
%O A210638 0,18
%A A210638 _Peter Luschny_, Mar 26 2012