A003713 -id:A003713 - cp's OEIS frontend

A000154 Erroneous version of A003713.

1, 1, 2, 7, 35, 228, 1834, 17382, 195866, 2487832, 35499576, 562356672, 9794156448, 186025364016, 3826961710272, 84775065603888, 2011929826983504

Offset: 0

dead

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

D. S. Kluk & N. J. A. Sloane, Correspondence, 1979

A089064 Expansion of e.g.f. log(1-log(1-x)).

Original entry on oeis.org

0, 1, 0, 1, 1, 8, 26, 194, 1142, 9736, 81384, 823392, 8738016, 104336880, 1328270880, 18419317968, 272291315376, 4312675967232, 72478365279360, 1292173575000192, 24314102888206464, 482046102448383744, 10037081891973037824

Offset: 0

Vladeta Jovovic, Dec 20 2003

Stirling transform of a(n)=[1,0,1,1,8,26,...] is A075792(n)=[1,1,2,8,44,...]. - Michael Somos, Mar 04 2004
Stirling transform of -(-1)^n*a(n)=[1,0,1,-1,8,-26,194,...] is A000142(n-1)=[1,1,2,6,24,120,...]. - Michael Somos, Mar 04 2004
Number of increasing trees on n vertices in which the second player has a winning strategy when interpreted as a game tree - Victor YZ Wang, May 03 2025
Convolution of absolute value of Mobius function and Mobius function applied to bottom and top elements of the set partition lattice - Victor YZ Wang, May 03 2025

G. H. Hardy, A Course of Pure Mathematics, 10th ed., Cambridge University Press, 1960, p. 428.

Seiichi Manyama, Table of n, a(n) for n = 0..451
G. H. Hardy, A Course of Pure Mathematics, Cambridge, The University Press, 1908.
Victor Wang On an alternating sum of factorials and Stirling numbers of the first kind: trees, lattices, and games, arXiv:2504.21176 [math.CO], 2025.

Cf. A003713, A075792.

nmax = 20; CoefficientList[Series[Log[1-Log[1-x]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jul 01 2018 *)
Table[(-1)^(n+1) * Sum[(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))

{a(n) = if (n<1, 0, (n-1)!-sum(k=1, n-1, binomial(n-1, k)*(k-1)!*a(n-k)))} \\ Seiichi Manyama, Jun 01 2019

a(n) = (-1)^(n+1)*Sum_{k=1..n} (k-1)!*Stirling1(n, k).
E.g.f.: log(1-log(1-x)).
a(n) = (n-1)! - Sum_{k=1..n-1} binomial(n-1,k) * (k-1)! * a(n-k). - Seiichi Manyama, Jun 01 2019

A039814 Matrix square of Stirling-1 triangle A008275.

Original entry on oeis.org

1, -2, 1, 7, -6, 1, -35, 40, -12, 1, 228, -315, 130, -20, 1, -1834, 2908, -1485, 320, -30, 1, 17582, -30989, 18508, -5005, 665, -42, 1, -195866, 375611, -253400, 81088, -13650, 1232, -56, 1, 2487832, -5112570, 3805723, -1389612, 279048, -32130, 2100, -72, 1

Offset: 1

Christian G. Bower, Feb 15 1999

Exponential Riordan array [1/((1 + x)*(1 + log(1 + x))), log(1 + log(1 + x))]. The row sums of the unsigned array give A007840 (apart from the initial term). - Peter Bala, Jul 22 2014

Also the Bell transform of (-1)^n*A003713(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 28 2016

			Triangle begins:
      1;
     -2,    1;
      7,   -6,     1;
    -35,   40,   -12,   1;
    228, -315,   130, -20,   1;
  -1834, 2908, -1485, 320, -30, 1;
...

Seiichi Manyama, Rows n = 1..140, flattened (Rows n = 1..60 from Vincenzo Librandi)

Column k=1..3 give (-1)^(n-1) * A003713(n), (-1)^n * A341587(n), (-1)^(n-1) * A341588(n).
Cf. A007840.
Cf. A039810, A039815, A039816, A039817.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> (-1)^n*add(k!*abs(Stirling1(n+1,k+1)), k=0..n), 10); # Peter Luschny, Jan 28 2016

max = 9; t = Table[StirlingS1[n, k], {n, 1, max}, {k, 1, max}]; t2 = t.t; Table[t2[[n, k]], {n, 1, max}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 01 2013 *)
rows = 9;
t = Table[(-1)^n*Sum[k!*Abs[StirlingS1[n+1, k+1]], {k,0,n}], {n, 0, rows}];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)

T(n, k) = sum(j=0, n, stirling(n, j, 1)*stirling(j, k, 1)); \\ Seiichi Manyama, Feb 13 2022

E.g.f. of k-th column: ((log(1+log(1+x)))^k)/k!.
E.g.f.: 1/(1 + t)*( 1 + log(1 + t) )^(x-1) = 1 + (-2 + x)*t + (7 - 6*x + x^2)*t^2/2! + .... - Peter Bala, Jul 22 2014
T(n,k) = Sum_{j=0..n} Stirling1(n,j) * Stirling1(j,k). - Seiichi Manyama, Feb 13 2022

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))))).

A000268 E.g.f.: -log(1+log(1+log(1-x))).

Original entry on oeis.org

1, 3, 15, 105, 947, 10472, 137337, 2085605, 36017472, 697407850, 14969626900, 352877606716, 9064191508018, 252024567201300, 7542036496650006, 241721880399970938, 8261159383595659128, 299916384730043070880, 11526945327529620432872, 467583770376898192016104

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).

Seiichi Manyama, Table of n, a(n) for n = 1..400 (terms 1..100 from T. D. Noe)
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 299
Kruchinin Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.

a(n)=|A039815(n, 1)| (first column of triangle).

Cf. A003713, A000310, A000359, A000406, A001765.

max = 20; CoefficientList[-Log[1 + Log[1 + Log[1 - x]]]/x + O[x]^max, x] * Range[max]! (* Jean-François Alcover, Feb 06 2016 *)

a(n) = sum(m=1, n, (m-1)!*(-1)^(n+m)*sum(k=m, n, stirling(n,k,1)*stirling(k,m,1))); \\ Michel Marcus, Feb 06 2016

a(n) = sum((m-1)!*(-1)^(n+m)*sum(stirling1(n, k)*stirling1(k, m), k,m,n), m,1,n), n>0. - Vladimir Kruchinin, Sep 14 2010

Revised description from Christian G. Bower, Aug 15 1998

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)))).

A341587 E.g.f.: log(1 + log(1 - x))^2 / 2.

Original entry on oeis.org

1, 6, 40, 315, 2908, 30989, 375611, 5112570, 77305024, 1286640410, 23387713930, 461187042992, 9808283703684, 223833267479764, 5456669750439788, 141540592345674800, 3892707724320135616, 113153294901088030320, 3466501398608272647984, 111636571036702743967104, 3770483138507706753943584

Offset: 2

Ilya Gutkovskiy, Feb 15 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 2..418

Cf. A000254, A000558, A003713, A008275, A039814, A052822, A302547, A302548, A341588.

nmax = 22; CoefficientList[Series[Log[1 + Log[1 - x]]^2/2, {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 2] &
Table[Sum[Abs[StirlingS1[n, k] StirlingS1[k, 2]], {k, 2, n}], {n, 2, 22}]

a(n) = Sum_{k=2..n} |Stirling1(n, k) * Stirling1(k, 2)|.
a(n) = Sum_{k=2..n} |Stirling1(n, k)| * (k-1)! * H(k-1), where H(k) is the k-th harmonic number.
a(n) = Sum_{k=1..n-1} binomial(n-1, k) * A003713(k) * A003713(n-k).
a(n) = A052822(n) / 2.
a(n) ~ sqrt(2*Pi) * log(n) * n^(n - 1/2) / (exp(1) - 1)^n * (1 + (gamma - log(exp(1) - 1))/log(n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 15 2021

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))))))).

A341588 E.g.f.: -log(1 + log(1 - x))^3 / 6.

Original entry on oeis.org

1, 12, 130, 1485, 18508, 253400, 3805723, 62437500, 1113510409, 21479997957, 446094038806, 9930796412082, 236037249893092, 5968192832899412, 160007282538148508, 4534905316824903144, 135500246340709682692, 4257646241716404353684, 140366073694357927723936, 4845119946789226304526392

Offset: 3

Ilya Gutkovskiy, Feb 15 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 3..418

Cf. A000399, A000559, A003713, A008275, A039814, A341587.

nmax = 22; CoefficientList[Series[-Log[1 + Log[1 - x]]^3/6, {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 3] &
Table[Sum[Abs[StirlingS1[n, k] StirlingS1[k, 3]], {k, 3, n}], {n, 3, 22}]

a(n) = Sum_{k=3..n} |Stirling1(n, k) * Stirling1(k, 3)|.

a(n) ~ (n-1)! * log(n)^2 / (2 * (1 - exp(-1))^n) * (1 + (2*gamma - 2*log(exp(1) - 1)) / log(n) + (gamma^2 - Pi^2/6 - 2*log(exp(1) - 1)*gamma + log(exp(1)-1)^2) / log(n)^2). - Vaclav Kotesovec, Jun 04 2022

cp's OEIS Frontend

A000154 Erroneous version of A003713.

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A089064 Expansion of e.g.f. log(1-log(1-x)).

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Mathematica

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A039814 Matrix square of Stirling-1 triangle A008275.

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Mathematica

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A000359 Coefficients of iterated exponentials.

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Mathematica

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PARI

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A000268 E.g.f.: -log(1+log(1+log(1-x))).

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Mathematica

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Extensions

A000310 Coefficients of iterated exponentials.

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Mathematica

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PARI

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A341587 E.g.f.: log(1 + log(1 - x))^2 / 2.

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A000406 Coefficients of iterated exponentials.

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