A144150 -id:A144150 - cp's OEIS frontend

A005946 Number of n-step mappings with 5 inputs.

1, 52, 358, 1304, 3455, 7556, 14532, 25488, 41709, 64660, 95986, 137512, 191243, 259364, 344240, 448416, 574617, 725748, 904894, 1115320, 1360471, 1643972, 1969628, 2341424, 2763525, 3240276, 3776202, 4376008, 5044579, 5786980, 6608456, 7514432, 8510513, 9602484

Offset: 1

N. J. A. Sloane

Hogg & Huberman paper has a misprint a(4)=304. - Sean A. Irvine, Oct 11 2016

T. Hogg and B. A. Huberman, Attractors on finite sets: the dissipative dynamics of computing structures, Phys. Review A 32 (1985), 2338-2346.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..10000
T. Hogg and B. A. Huberman, Attractors on finite sets: the dissipative dynamics of computing structures, Phys. Review A 32 (1985), 2338-2346. (Annotated scanned copy)
B. A. Huberman, T. H. Hogg, & N. J. A. Sloane, Correspondence, 1985
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Row n=5 of A144150.

b:= proc(n, k) option remember; `if`(k=0, `if`(n<2, 1, 0),
      add(Stirling2(n, j)*b(j, k-1), j=0..n))
    end:
a:= n-> b(5, n):
seq(a(n), n=1..36);  # Alois P. Heinz, Aug 23 2021

LinearRecurrence[{5, -10, 10, -5, 1}, {1, 52, 358, 1304, 3455}, 36] (* Jean-François Alcover, May 20 2022 *)

a(n) = h(5,n) where h(n, m) = Sum_{j} (n!/f(j)) * Product_{k=1..n} h(k,m-1)^(j(k)) and the sum runs over all partitions j=(j(1),...,j(n)) of n and f(j) = Product_{k=1..n} j(k)! * (k!)^(j(k)). That is, j satisfies Sum_{k=1..n} k*j(k) = n [From Hogg & Huberman]. - Sean A. Irvine, Oct 11 2016

G.f.: x*(24*x^3+108*x^2+47*x+1)/(1-x)^5. - Alois P. Heinz, Aug 23 2021

a(4) corrected and more terms from Sean A. Irvine, Oct 11 2016

A209631 Square array A(n,k), n>=0, k>=0, read by antidiagonals, A(n,k) = exponential transform applied n times to identity function, evaluated at k.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 10, 4, 1, 1, 5, 20, 41, 5, 1, 1, 6, 33, 127, 196, 6, 1, 1, 7, 49, 280, 967, 1057, 7, 1, 1, 8, 68, 518, 2883, 8549, 6322, 8, 1, 1, 9, 90, 859, 6689, 34817, 85829, 41393, 9, 1, 1, 10, 115, 1321, 13310, 101841, 481477

Offset: 0

Peter Luschny, Mar 11 2012

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

			n\k [0][1][2] [3]   [4]    [5]     [6]
[0]  0, 1, 2,  3,    4,     5,      6
[1]  1, 1, 3, 10,   41,   196,   1057   [A000248]
[2]  1, 1, 4, 20,  127,   967,   8549   [A007550]
[3]  1, 1, 5, 33,  280,  2883,  34817
[4]  1, 1, 6, 49,  518,  6689, 101841
[5]  1, 1, 7, 68,  859, 13310, 243946
[6]  1, 1, 8, 90, 1321, 23851, 510502
column3(n) = (3*n^2 + 11*n + 6)/2!
column4(n) = (18*n^3 + 93*n^2 + 111*n + 24)/3!
column5(n) = (180*n^4 + 1180*n^3 + 2160*n^2 + 1064*n + 120)/4!
column6(n) = (2700*n^5+21225*n^4+51850*n^3+41835*n^2+8510*n+720)/5!

Discussion on seqcomp: A little challenge

Cf. A000248, A007550, A111672, A144150.

# Implementation after Alois P. Heinz.
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:
A209631 := (n,k) -> (exptr@@n)(m->m)(k):
seq(lprint(seq(A209631(n,k), k=0..6)), n=0..6);

exptr[p_] := Module[{g}, g[n_] := g[n] = Module[{k}, If[n == 0, 1, Sum[Binomial[n-1, k-1]*p[k]*g[n-k], {k, 1, n}]]]; g]; A209631[n_, k_] := Nest[exptr, Identity, n][k]; Table[A209631[n-k , k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 27 2014, after Alois P. Heinz *)

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.

Original entry on oeis.org

-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

Peter Luschny, Mar 26 2012

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).

			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

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.

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.

Cf. A000587, A144150, A209631.

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);

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 *)

A381931 Triangular array T(n, k) read by rows: denominators of the coefficients for the iterated exponential F^{r}(x) = x + Sum_{n>=1} x^(n+1)Sum_{k=1..n} r^(n+1-k)A381932(n, k)/T(n, k) with F^{1}(x) = exp(x)-1 and F^{2}(x) = exp(exp(x)-1)-1.

Original entry on oeis.org

2, 4, 12, 8, 48, 48, 16, 144, 24, 180, 32, 1152, 1728, 5760, 8640, 64, 640, 3456, 5760, 17280, 6720, 128, 7680, 34560, 1152, 34560, 32256, 241920, 256, 26880, 82944, 414720, 41472, 580608, 107520, 1451520, 512, 430080, 645120, 622080, 4147200, 6967296, 21772800, 87091200, 43545600

Offset: 1

Thomas Scheuerle, Mar 10 2025

This is the main entry for this sequence of fractions.
Convergence and analytic continuation of this series representation are interesting research topics with many unsolved problems and open questions.
Evaluating the polynomial of row n P(x) = Sum_{k=1..n} x^(n+1-k)*A381932(n, k)/T(n, k) gives A144150(n+1, x-1)/(n+1)!.

			Triangle T(n, k) begins:
[1]  2;
[2]  4,   12;
[3]  8,   48,     48;
[4]  16,  144,    24,     180;
[5]  32,  1152,   1728,   5760,   8640;
[6]  64,  640,    3456,   5760,   17280,   6720;
[7]  128, 7680,   34560,  1152,   34560,   32256,   241920;
[8]  256, 26880,  82944,  414720, 41472,   580608,  107520,   1451520;
[9]  512, 430080, 645120, 622080, 4147200, 6967296, 21772800, 87091200, 43545600;
.
F^{r}(x) = x
+ x^2*1/2*r
+ x^3*(1/4*r^2 - 1/12*r)
+ x^4*(1/8*r^3 - 5/48*r^2 + 1/48*r)
+ x^5*(1/16*r^4 - 13/144*r^3 + 1/24*r^2 - 1/180*r)
+ x^6*(1/32*r^5 - 77/1152*r^4 + 89/1728*r^3 - 91/5760*r^2 + 11/8640*r)
+ ... .

Gottfried Helms, Coefficients for fractional iterates exp(x)-1
MathOverflow, f(f(x)) = exp(x)-1 and other functions "just in the middle" between linear and exponential

Cf. A381932 (numerators).

Cf. A052123, A052122, A052104, A052105, A144150, A180609, A184011.

c(k, n) = {my(f=x); for(m=1, k, f=subst(f, x, exp(x)-1)); polcoeff(f+O(x^(n+1)), n)}
row(n) = my(p=polinterpolate(vector(2*(n+1), k, k-1), vector(2*(n+1), k, c(k-1, n+1)))); vector(n, k, denominator(polcoeff(p, n-k+1)));

T(n, 1) = 2^n.

T(n, n) = denominator(A180609(n)/(n!*(n+1)!)).

A381932 Triangular array T(n, k) read by rows: denominators of the coefficients for the iterated exponential F^{r}(x) = x + Sum_{n>=1} x^(n+1)Sum_{k=1..n} r^(n+1-k)T(n, k)/A381931(n, k) with F^{1}(x) = exp(x)-1 and F^{2}(x) = exp(exp(x)-1)-1.

Original entry on oeis.org

1, 1, -1, 1, -5, 1, 1, -13, 1, -1, 1, -77, 89, -91, 11, 1, -29, 175, -149, 91, -1, 1, -223, 1501, -37, 391, -43, -11, 1, -481, 2821, -13943, 725, -2357, 17, 29, 1, -4609, 16099, -19481, 91313, -55649, 23137, 1727, 493, 1, -4861, 89993, -933293, 399637, -1061231, 2035739, -8189, 4897, -2711

Offset: 1

Thomas Scheuerle, Mar 12 2025

The main entry for this sequence of fractions is in A381931.

			Triangle T(n, k) begins:
[1]  1;
[2]  1,    -1;
[3]  1,    -5,     1;
[4]  1,   -13,     1,     -1;
[5]  1,   -77,    89,    -91,    11;
[6]  1,   -29,   175,   -149,    91,     -1;
[7]  1,  -223,  1501,    -37,   391,    -43,   -11;
[8]  1,  -481,  2821, -13943,   725,  -2357,    17,   29;
[9]  1, -4609, 16099, -19481, 91313, -55649, 23137, 1727, 493;
.
F^{r}(x) = x
+ x^2*1/2*r
+ x^3*(1/4*r^2 - 1/12*r)
+ x^4*(1/8*r^3 - 5/48*r^2 + 1/48*r)
+ x^5*(1/16*r^4 - 13/144*r^3 + 1/24*r^2 - 1/180*r)
+ x^6*(1/32*r^5 - 77/1152*r^4 + 89/1728*r^3 - 91/5760*r^2 + 11/8640*r)
+ ... .

Cf. A381931 (denominators).
Cf. A052123, A052122, A052104, A052105
Cf. A064169, A144150, A180609, A184011.

c(k, n) = {my(f=x); for(m=1, k, f=subst(f, x, exp(x)-1)); polcoeff(f+O(x^(n+1)), n)}
row(n) = my(p=polinterpolate(vector(2*(n+1), k, k-1), vector(2*(n+1), k, c(k-1, n+1)))); vector(n, k, numerator(polcoeff(p, n-k+1)));

Conjecture: abs(T(n, 2)) = A064169(n - 1).

T(n, n) = numerator(A180609(n)/(n!*(n+1)!)).

cp's OEIS Frontend

A005946 Number of n-step mappings with 5 inputs.

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A209631 Square array A(n,k), n>=0, k>=0, read by antidiagonals, A(n,k) = exponential transform applied n times to identity function, evaluated at k.

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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.

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A381931 Triangular array T(n, k) read by rows: denominators of the coefficients for the iterated exponential F^{r}(x) = x + Sum_{n>=1} x^(n+1)*Sum_{k=1..n} r^(n+1-k)*A381932(n, k)/T(n, k) with F^{1}(x) = exp(x)-1 and F^{2}(x) = exp(exp(x)-1)-1.

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A381932 Triangular array T(n, k) read by rows: denominators of the coefficients for the iterated exponential F^{r}(x) = x + Sum_{n>=1} x^(n+1)*Sum_{k=1..n} r^(n+1-k)*T(n, k)/A381931(n, k) with F^{1}(x) = exp(x)-1 and F^{2}(x) = exp(exp(x)-1)-1.

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A381931 Triangular array T(n, k) read by rows: denominators of the coefficients for the iterated exponential F^{r}(x) = x + Sum_{n>=1} x^(n+1)Sum_{k=1..n} r^(n+1-k)A381932(n, k)/T(n, k) with F^{1}(x) = exp(x)-1 and F^{2}(x) = exp(exp(x)-1)-1.

A381932 Triangular array T(n, k) read by rows: denominators of the coefficients for the iterated exponential F^{r}(x) = x + Sum_{n>=1} x^(n+1)Sum_{k=1..n} r^(n+1-k)T(n, k)/A381931(n, k) with F^{1}(x) = exp(x)-1 and F^{2}(x) = exp(exp(x)-1)-1.