A000268 -id:A000268 - cp's OEIS frontend

A003713 Expansion of e.g.f. log(1/(1+log(1-x))).

0, 1, 2, 7, 35, 228, 1834, 17582, 195866, 2487832, 35499576, 562356672, 9794156448, 186025364016, 3826961710272, 84775065603888, 2011929826983504, 50929108873336320, 1369732445916318336, 39005083331889816960, 1172419218038422659456, 37095226237402478348544

Offset: 0

N. J. A. Sloane, R. H. Hardin, Simon Plouffe

a(n+1) is the permanent of the n X n matrix M with M(i,i) = i+1, other entries 1. - Philippe Deléham, Nov 03 2005
Supernecklaces of type III (cycles of cycles). - Ricardo Bittencourt, May 05 2013
Unsigned coefficients for the raising / creation operator R for the Appell sequence of polynomials A238385: R = x + 1 - 2 D + 7 D^2/2! - 35 D^3/3! + ... . - Tom Copeland, May 09 2016

J. Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 125.
Jekuthiel Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353. [Annotated scanned copy]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 34
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 298

a(n)=|A039814(n, 1)| (first column of triangle). Cf. A000268, A000310, A000359, A000406, A001765.
Cf. A007840, A089064.
Cf. A238385.

series(ln(1/(1+ln(1-x))),x,17);
with (combstruct): M[ 1798 ] := [ A,{A=Cycle(Cycle(Z))},labeled ]:

With[{nn=20},CoefficientList[Series[Log[1/(1+Log[1-x])],{x,0,nn}],x]Range[0,nn]!] (* Harvey P. Dale, Dec 15 2012 *)
Table[Sum[(-1)^(n-k) * (k-1)! * StirlingS1[n, k], {k, 1, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 19 2024 *)

a(n)=if(n<0,0,n!*polcoeff(-log(1+log(1-x+x*O(x^n))),n))

Sum_{k=1..n} (k-1)!*|Stirling1(n, k)|. - Vladeta Jovovic, Sep 14 2003
a(n+1) = n! * Sum_{k=0..n} A007840(k)/k!. E.g., a(4) = 228 = 24*(1/1 + 1/1 + 3/2 + 14/6 + 88/24) = 24 + 24 + 36 + 56 + 88. - Philippe Deléham, Dec 10 2003
a(n) ~ (n-1)! * (exp(1)/(exp(1)-1))^n. - Vaclav Kotesovec, Jun 21 2013
a(0) = 0; a(n) = (n-1)! + Sum_{k=1..n-1} binomial(n-1,k) * (k-1)! * a(n-k). - Ilya Gutkovskiy, Jul 18 2020

Thanks to Paul Zimmermann for comments.

A000359 Coefficients of iterated exponentials.

Original entry on oeis.org

1, 5, 40, 440, 6170, 105315, 2120610, 49242470, 1296133195, 38152216495, 1242274374380, 44345089721923, 1722416374173854, 72330102999829054, 3265871028909088036, 157797437377747327987, 8124524883679977475839, 444098724261935142753430

Offset: 1

N. J. A. Sloane

nonn

J. Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..100
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Jekuthiel Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353. [Annotated scanned copy]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 301

a(n) = |A039817(n, 1)| (first column of triangle). Cf. A003713, A000268, A000310, A000406, A001765.

max = 20; CoefficientList[-Log[1 + Log[1 + Log[1 + Log[1 + Log[1 - x]]]]]/x + O[x]^max, x]*Range[max]! (* Jean-François Alcover, Feb 08 2016 *)

T(n, k) = if(k==1, (n-1)!, sum(j=1, n, abs(stirling(n, j, 1))*T(j, k-1)));
a(n) = T(n, 5); \\ Seiichi Manyama, Feb 11 2022

my(N=20, x='x+O('x^N)); Vec(serlaplace(-log(1+log(1+log(1+log(1+log(1-x))))))) \\ Seiichi Manyama, Feb 11 2022

E.g.f.: -log(1+log(1+log(1+log(1+log(1-x))))).

A000310 Coefficients of iterated exponentials.

Original entry on oeis.org

1, 4, 26, 234, 2696, 37919, 630521, 12111114, 264051201, 6445170229, 174183891471, 5164718385337, 166737090160871, 5822980248613990, 218756388226681557, 8797723991458469015, 377159237609540937788, 17170729962232112834302, 827382365085791968518198, 42070004707327023844695198

Offset: 1

N. J. A. Sloane

nonn

J. Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..100
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Jekuthiel Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353. [Annotated scanned copy]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 300

a(n) = |A039816(n, 1)| (first column of triangle). Cf. A003713, A000268, A000359, A000406, A001765.

max = 20; CoefficientList[-Log[1 + Log[1 + Log[1 + Log[1 - x]]]]/x + O[x]^max, x]*Range[max]! (* Jean-François Alcover, Feb 07 2016 *)

T(n, k) = if(k==1, (n-1)!, sum(j=1, n, abs(stirling(n, j, 1))*T(j, k-1)));
a(n) = T(n, 4); \\ Seiichi Manyama, Feb 11 2022

my(x='x+O('x^40)); Vec(serlaplace(-log(1+log(1+log(1+log(1-x)))))); \\ Michel Marcus, Feb 11 2022

E.g.f.: -log(1+log(1+log(1+log(1-x)))).

A000406 Coefficients of iterated exponentials.

Original entry on oeis.org

1, 6, 57, 741, 12244, 245755, 5809875, 158198200, 4877852505, 168055077875, 6400217406500, 267058149580823, 12118701719205803, 594291742526530761, 31323687504696772151, 1766116437541895988303, 106080070002238888908150

Offset: 1

N. J. A. Sloane

nonn

J. Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..100
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Jekuthiel Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353. [Annotated scanned copy]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 302

Cf. A003713, A000268, A000310, A000359, A001765.

With[{nn=20},CoefficientList[Series[-Log[1+Log[1+Log[1+Log[1+ Log[1+ Log[1- x]]]]]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Nov 03 2015 *)

T(n, k) = if(k==1, (n-1)!, sum(j=1, n, abs(stirling(n, j, 1))*T(j, k-1)));
a(n) = T(n, 6); \\ Seiichi Manyama, Feb 11 2022

my(N=20, x='x+O('x^N)); Vec(serlaplace(-log(1+log(1+log(1+log(1+log(1+log(1-x)))))))) \\ Seiichi Manyama, Feb 11 2022

E.g.f.: -log(1+log(1+log(1+log(1+log(1+log(1-x)))))).

A001765 Coefficients of iterated exponentials.

Original entry on oeis.org

1, 7, 77, 1155, 21973, 506989, 13761937, 429853851, 15192078027, 599551077881, 26140497946017, 1248134313062231, 64783855286002573, 3632510833677434324, 218845138322691595694, 14099918095287618382033, 967508237903439910445565, 70447525748137979196484589

Offset: 1

N. J. A. Sloane

J. Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353.
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, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..100
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
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)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 303

Cf. A003713, A000268, A000310, A000359, A000406.

With[{nn=20},CoefficientList[Series[-Log[1+Log[1+Log[1+Log[1+Log[1+Log[1+Log[1-x]]]]]]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jan 07 2023 *)

T(n, k) = if(k==1, (n-1)!, sum(j=1, n, abs(stirling(n, j, 1))*T(j, k-1)));
a(n) = T(n, 7); \\ Seiichi Manyama, Feb 11 2022

my(N=20, x='x+O('x^N)); Vec(serlaplace(-log(1+log(1+log(1+log(1+log(1+log(1+log(1-x))))))))) \\ Seiichi Manyama, Feb 11 2022

E.g.f.: -log(1+log(1+log(1+log(1+log(1+log(1+log(1-x))))))).

A039815 Triangle read by rows: matrix cube of the Stirling-1 triangle A008275.

Original entry on oeis.org

1, -3, 1, 15, -9, 1, -105, 87, -18, 1, 947, -975, 285, -30, 1, -10472, 12657, -4680, 705, -45, 1, 137337, -188090, 82887, -15960, 1470, -63, 1, -2085605, 3159699, -1598954, 370237, -43890, 2730, -84, 1, 36017472, -59326371, 33613353, -9009294, 1292067, -103950, 4662, -108, 1

Offset: 1

Christian G. Bower, Feb 15 1999

			Triangle begins:
       1;
      -3,     1;
      15,    -9,     1;
    -105,    87,   -18,   1;
     947,  -975,   285, -30,   1;
  -10472, 12657, -4680, 705, -45, 1;
  ...

Seiichi Manyama, Rows n = 1..140, flattened
Gabriella Bretti, Pierpaolo Natalini and Paolo E. Ricci, A new set of Sheffer-Bell polynomials and logarithmic numbers, Georgian Mathematical Journal, Feb. 2019, page 9.

Cf. A000268 (first column), A008275.

Cf. A039811, A039814, A039816, A039817.

T:= Matrix(10,10,(i,j) -> `if`(i>= j, combinat:-stirling1(i,j),0)):
M:= T^3:
seq(seq(M[i,j],j=1..i),i=1..10); # Robert Israel, Sep 12 2022

Flatten[Table[SeriesCoefficient[(Log[1+Log[1+Log[1+x]]])^k, {x,0,n}] n!/k!, {n,9}, {k,n}]] (* Stefano Spezia, Sep 12 2022 *)

E.g.f. of k-th column: ((log(1+log(1+log(1+x))))^k)/k!.

A053577 Cototient function n - phi(n) is a power of 2.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 11, 12, 13, 14, 16, 17, 19, 23, 24, 28, 29, 31, 32, 37, 41, 43, 47, 48, 53, 56, 59, 61, 62, 64, 67, 71, 73, 79, 83, 89, 96, 97, 101, 103, 107, 109, 112, 113, 124, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 192, 193, 197

Offset: 1

Labos Elemer, Jan 18 2000

nonn

Consists of all primes, powers of 2, and powers of 2 times Mersenne primes (A000268). - Robert Israel, Jan 29 2018

			For p prime, cototient(p)=1; for x in the set {49152,57344,63488,65024,65528,65536}, x-phi(x) = 2^15 = 32768.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000010, A000268, A051953, A053578.

filter:= proc(n) local x; x:= n - numtheory:-phi(n); x = 2^padic:-ordp(x,2) end proc:
select(filter, [$1..300]); # Robert Israel, Jan 29 2018

Select[Range[200],IntegerQ[Log[2,#-EulerPhi[#]]]&] (* Harvey P. Dale, Dec 14 2011 *)

select(x->hammingweight(x-eulerphi(x))==1,[1..200]) \\ M. F. Hasler, Nov 10 2016

A051953(n) = 2^w.

Edited by M. F. Hasler, Nov 10 2016

A302358 a(n) = coefficient of x^n in the n-th iteration (n-fold self-composition) of e.g.f. -log(1 - x).

Original entry on oeis.org

1, 2, 15, 234, 6170, 245755, 13761937, 1030431500, 99399019626, 12003835242090, 1773907219147800, 314880916127332489, 66109411013740671200, 16204039283106534720952, 4585484528618722750937783, 1483746673734716952089913364, 544359300175753347889146067840

Offset: 1

Ilya Gutkovskiy, Apr 06 2018

nonn

			The initial coefficients of successive iterations of e.g.f. A(x) = -log(1 - x) are as follows:
n = 1: 0, (1), 1,   2,    6,    24,  ... e.g.f. A(x)
n = 2: 0,  1, (2),  7,   35,   228,  ... e.g.f. A(A(x))
n = 3: 0,  1,  3, (15), 105,   947,  ... e.g.f. A(A(A(x)))
n = 4: 0,  1,  4,  26, (234), 2696,  ... e.g.f. A(A(A(A(x))))
n = 5: 0,  1,  5,  40,  440, (6170), ... e.g.f. A(A(A(A(A(x)))))

Seiichi Manyama, Table of n, a(n) for n = 1..246
Jekuthiel Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353. [Annotated scanned copy]
N. J. A. Sloane, Transforms

Cf. A000268, A000310, A000359, A000406, A001765, A003713, A104150, A139383, A158832, A174482, A261280.

g:= x-> -log(1-x):
a:= n-> n! * coeff(series((g@@n)(x), x, n+1), x, n):
seq(a(n), n=1..19);  # Alois P. Heinz, Feb 11 2022

Table[n! SeriesCoefficient[Nest[Function[x, -Log[1 - x]], x, n], {x, 0, n}], {n, 17}]

T(n, k) = if(k==1, (n-1)!, sum(j=1, n, abs(stirling(n, j, 1))*T(j, k-1)));
a(n) = T(n, n); \\ Seiichi Manyama, Feb 11 2022

a(n) = T(n,n), T(n,k) = Sum_{j=1..n} |Stirling1(n,j)| * T(j,k-1), k>1, T(n,1) = (n-1)!. - Seiichi Manyama, Feb 11 2022

A111933 Triangle read by rows, generated from Stirling cycle numbers.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 7, 6, 1, 4, 15, 35, 24, 1, 5, 26, 105, 228, 120, 1, 6, 40, 234, 947, 1834, 720, 1, 7, 57, 440, 2696, 10472, 17582, 5040, 1, 8, 77, 741, 6170, 37919, 137337, 195866, 40320, 1, 9, 100, 1155, 12244, 105315, 630521, 2085605, 2487832, 362880

Offset: 1

Gary W. Adamson, Aug 21 2005

Let M = the infinite lower triangular matrix of Stirling cycle numbers (A008275). Perform M^n * [1, 0, 0, 0, ...] forming an array. Antidiagonals of that array become the rows of this triangle.

			Row 5 of the triangle = 1, 4, 15, 35, 24; generated from M^n * [1,0,0,0,...] (n = 1 through 5); then take antidiagonals.
Terms in the array, first few rows are:
  1, 1,  2,   6,    24,    120, ...
  1, 2,  7,  35,   228,   1834, ...
  1, 3, 15, 105,   947,  10472, ...
  1, 4, 26, 234,  2697,  37919, ...
  1, 5, 40, 440,  6170, 105315, ...
  1, 6, 57, 741, 12244, 245755, ...
  ...
First few rows of the triangle are:
  1;
  1, 1;
  1, 2,  2;
  1, 3,  7,   6;
  1, 4, 15,  35,  24;
  1, 5, 26, 105, 228,  120;
  1, 6, 40, 234, 947, 1834, 720;
  ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Row 1..7 give A000142(n-1), A003713, A000268, A000310, A000359, A000406, A001765.
Column 3 of the array = A005449.
Column 4 of the array = A094952.
Cf. A008275, A302358.

a(28), a(36) and a(45) corrected by Seiichi Manyama, Feb 11 2022

A351525 Expansion of e.g.f. (log(1 + log(1 + log(1+ x))))^2 / 2.

Original entry on oeis.org

1, -9, 87, -975, 12657, -188090, 3159699, -59326371, 1232843529, -28116615263, 698649506871, -18796044698977, 544507930693022, -16903759793180115, 559960766050363931, -19719027513960290370, 735696883534117583082, -28991986984973263419262

Offset: 2

Seiichi Manyama, Feb 13 2022

sign

Column 2 of A039815.

Cf. A000268, A341587, A351513, A351526, A351527.

With[{nn=20},CoefficientList[Series[Log[1+Log[1+Log[1+x]]]^2/2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jan 15 2024 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(log(1+log(1+log(1+x)))^2/2))

T(n, k) = if(k==0, n==1, sum(j=0, n, abs(stirling(n, j, 1))*T(j, k-1)));
a(n) = (-1)^n*sum(k=1, n-1, binomial(n-1, k)*T(k, 3)*T(n-k, 3));

a(n) = (-1)^n * Sum_{k=1..n-1} binomial(n-1,k) * A000268(k) * A000268(n-k).

cp's OEIS Frontend

A003713 Expansion of e.g.f. log(1/(1+log(1-x))).

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A000359 Coefficients of iterated exponentials.

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A000310 Coefficients of iterated exponentials.

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A000406 Coefficients of iterated exponentials.

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A001765 Coefficients of iterated exponentials.

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A039815 Triangle read by rows: matrix cube of the Stirling-1 triangle A008275.

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A053577 Cototient function n - phi(n) is a power of 2.

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