A003704 -id:A003704 - cp's OEIS frontend

A202139 Expansion of e.g.f. log(1/(1-artanh(x))).

0, 1, 1, 4, 14, 88, 544, 4688, 41712, 459520, 5333376, 71876352, 1027670016, 16428530688, 278818065408, 5167215464448, 101437811718144, 2140879726411776, 47698275298050048, 1130276555155243008, 28167446673847812096, 740796870212763254784

Offset: 0

Vladimir Kruchinin, Dec 12 2011

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..431
Wikipedia contributors, Area function (inverse hyperbolic function), Wikipedia, the free encyclopedia, as of April 7, 2025.

Cf. A003704, A226968.

With[{nn=30},CoefficientList[Series[Log[1/(1-ArcTanh[x])],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Sep 10 2022 *)

a(n):=n!*sum(((m-1)!*sum((stirling1(k+m,m)*2^k*binomial(n-1,k+m-1))/(k+m)!,k,0,n-m)),m,1,n);

a_vector(n) = my(v=vector(n+1)); v[1]=0; for(i=1, n, v[i+1]=(i%2)*(i-1)!+sum(j=1, i\2, (2*j-2)!*binomial(i-1, 2*j-1)*v[i-2*j+2])); v; \\ Seiichi Manyama, Apr 30 2022

a(n) = n! * Sum_{m=1..n} (m-1)! * Sum_{k=0..n-m} Stirling1(k+m,m) * 2^k * binomial(n-1,k+m-1)/(k+m)!.
E.g.f.: log(2) - log(2 + log((1-x)/(1+x))). - Arkadiusz Wesolowski, Feb 19 2013
a(n) ~ n! * ((exp(2)+1)/(exp(2)-1))^n/n. - Vaclav Kotesovec, Jun 13 2013
a(0) = 0; a(n) = (n mod 2) * (n-1)! + Sum_{k=1..floor(n/2)} (2*k-2)! * binomial(n-1,2*k-1) * a(n-2*k+1). - Seiichi Manyama, Apr 30 2022

Zero prepended by Harvey P. Dale, Sep 10 2022

A296435 Expansion of e.g.f. log(1 + arcsinh(x)).

Original entry on oeis.org

0, 1, -1, 1, -2, 13, -64, 173, -720, 12409, -114816, 370137, -1491456, 88556037, -1263184896, 2668274373, 21448022016, 2491377242481, -50233550831616, -34526890553679, 5153298175033344, 202383113207336829, -5453228045913292800, -25792743610973373219, 1393299559788718325760

Offset: 0

Ilya Gutkovskiy, Dec 12 2017

sign

			E.g.f.: A(x) = x/1! - x^2/2! + x^3/3! - 2*x^4/4! + 13*x^5/5! - 64*x^6/6! + ...

Robert Israel, Table of n, a(n) for n = 0..451

Cf. A001818, A003704, A110708, A189815, A202139, A296437.

S:= series(ln(1+arcsinh(x)),x,51):
seq(coeff(S,x,j)*j!,j=0..50); # Robert Israel, Dec 12 2017

nmax = 24; CoefficientList[Series[Log[1 + ArcSinh[x]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 24; CoefficientList[Series[Log[1 + Log[x + Sqrt[1 + x^2]]], {x, 0, nmax}], x] Range[0, nmax]!

Vecrev(Pol(serlaplace(log(1 + asinh(x + O(x^30)))))) \\ Andrew Howroyd, Dec 12 2017

E.g.f.: log(1 + log(x + sqrt(1 + x^2))).

a(n) ~ 4*(Pi*cos(Pi*n/2) + 2*sin(Pi*n/2)) * n^(n-1) / ((4 + Pi^2) * exp(n)). - Vaclav Kotesovec, Dec 21 2017

A354056 Product_{n>=1} 1 / (1 - x^n)^(a(n)/n!) = 1 + sinh(x).

Original entry on oeis.org

1, -2, 1, -4, 21, -196, 1023, -5440, 65145, -1237456, 10925883, -69882880, 1994183205, -39099282496, 372390766023, -6270496768000, 158096182329585, -3268815510804736, 64115697136312563, -1009052458754375680, 27389518837925527965, -924645800211698308096, 19391677044464348893503

Offset: 1

Ilya Gutkovskiy, May 16 2022

sign

Cf. A003704, A353608, A353910, A354055, A354063, A354064, A354065, A354066.

nmax = 23; CoefficientList[Series[Sum[MoebiusMu[k] Log[1 + Sinh[x^k]]/k, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest

E.g.f.: Sum_{k>=1} mu(k) * log(1 + sinh(x^k)) / k.

A012494 Expansion of e.g.f. arctan(sin(x)) (odd powers only).

Original entry on oeis.org

1, -3, 45, -1743, 125625, -14554683, 2473184805, -579439207623, 179018972217585, -70518070842040563, 34495620120141463965, -20515677772241956573503, 14578232896601243652363945, -12198268199871431840616166443, 11871344562637111570703016357525

Offset: 0

Patrick Demichel (patrick.demichel(AT)hp.com)

arctan(sin(x)) = x - 3*x^3/3! + 45*x^5/5! - 1743*x^7/7! + 125625*x^9/9! + ....
Absolute values are coefficients in expansion of
arctanh(arcsinh(x)) = x + 3*x^3/3! + 45*x^5/5! + 1743*x^7/7! + ....
arccot(sin(x)) = Pi/2 - x + 3*x^3/3! - 45*x^5/5! + 1743*x^7/7! - ....

Vincenzo Librandi, Table of n, a(n) for n = 0..127

Bisection of A003704, A013208.
Cf. A101923, A001209, A000364, A000281, A156134, A002437.
Cf. other sequences with a g.f. of the form cos(x)/(1 - k*sin^2(x)): A000364 (k=1), A001209 (k=1/2), A000281 (k=2), A156134 (k=3), A002437 (k=4).

a:= n-> (t-> t!*coeff(series(arctan(sin(x)), x, t+1), x, t))(2*n+1):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 16 2018

Drop[ Range[0, 25]! CoefficientList[ Series[ ArcTan[ Sin[x]], {x, 0, 25}], x], {1, 25, 2}] (* Or *)
f[n_] := n!Sum[(1 + (-1)^(n - 2k + 1))2^(1 - 2k)Sum[(-1)^((n + 1)/2 - j)Binomial[2k - 1, j]((2j - 2k + 1)^n/n!)/(2k - 1), {j, 0, (2k - 1)/2}], {k, Ceiling[n/2]}]; Table[ f[n], {n, 1, 25, 2}] (* Robert G. Wilson v *)

a(n):=n!*sum((1+(-1)^(n-2*k+1))*2^(1-2*k)*sum((-1)^((n+1)/2-i)*binomial(2*k-1,i)*(2*i-2*k+1)^n/n!,i,0,(2*k-1)/2)/(2*k-1),k,1,ceiling((n)/2)); /* Vladimir Kruchinin, Feb 25 2011 */

a(n):=sum(sum((2*i-2*k-1)^(2*n+1)*binomial(2*k+1,i)*(-1)^(n-i+1),i,0,k)/(4^k*(2*k+1)),k,0,n); /* Vladimir Kruchinin, Feb 04 2012 */

a(n) = n!*sum(k=1..ceiling(n/2), (1+(-1)^(n-2*k+1))*2^(1-2*k)*sum(i=0..(2*k-1)/2, (-1)^((n+1)/2-i)*binomial(2*k-1,i)*(2*i-2*k+1)^n/n!)/(2*k-1)), n>0. Vladimir Kruchinin, Feb 25 2011
G.f.: cos(x) /(1 + sin^2(x)) = 1 - 3*x^2/2! + 45*x^4/4! - ... . - Peter Bala, Feb 06 2017
a(n) ~ (-1)^n * (2*n)! / (log(1+sqrt(2)))^(2*n+1). - Vaclav Kotesovec, Aug 17 2018

A331608 E.g.f.: exp(1 / (1 - sinh(x)) - 1).

Original entry on oeis.org

1, 1, 3, 14, 85, 632, 5559, 56352, 645929, 8252352, 116189291, 1786361216, 29764770941, 534082233856, 10264484355103, 210312181051392, 4575364233983057, 105310034714202112, 2556360647841415379, 65261358332774277120, 1747713179543456515749

Offset: 0

Ilya Gutkovskiy, Jan 22 2020

nonn

Cf. A003704, A003724, A006154, A075729, A331607, A331611.

nmax = 20; CoefficientList[Series[Exp[1/(1 - Sinh[x]) - 1], {x, 0, nmax}], x] Range[0, nmax]!
A006154[n_] := Sum[Sum[(-1)^j (k - 2 j)^n Binomial[k, j]/2^k, {j, 0, k}], {k, 1, n}]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] A006154[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A006154(k) * a(n-k).

a(n) ~ exp(1/(2^(3/2) * log(1 + sqrt(2))) - 3/4 + 2^(3/4) * sqrt(n) / sqrt(log(1 + sqrt(2))) - n) * n^(n - 1/4) / (2^(5/8) * log(1 + sqrt(2))^(n + 1/4)). - Vaclav Kotesovec, Jan 27 2020

A274805 The logarithmic transform of sigma(n).

Original entry on oeis.org

1, 2, -3, -6, 45, 11, -1372, 4298, 59244, -573463, -2432023, 75984243, -136498141, -10881169822, 100704750342, 1514280063802, -36086469752977, -102642110690866, 11883894518252419, -77863424962770751, -3705485804176583500, 71306510264347489177

Offset: 1

Johannes W. Meijer, Jul 27 2016

sign

The logarithmic transform [LOG] transforms an input sequence b(n) into the output sequence a(n). The LOG transform is the inverse of the exponential transform [EXP], see the Weisstein link and the Sloane and Plouffe reference. This relation goes by the name of Riddell’s formula. For information about the EXP transform see A274804. The logarithmic transform is related to the inverse multinomial transform, see A274844 and the first formula.
The definition of the LOG transform, see the second formula, shows that n >= 1. To preserve the identity EXP[LOG[b(n)]] = b(n) for n >= 0 for a sequence b(n) with offset 0 the shifted sequence b(n-1) with offset 1 has to be used as input for the LOG transform, otherwise information about b(0) will be lost in transformation.
In the a(n) formulas, see the examples, the cumulant expansion numbers A127671 appear.
We observe that the logarithmic transform leaves the value of a(0) undefined.
The Maple programs can be used to generate the logarithmic transform of a sequence. The first program uses a formula found by Alois P. Heinz, see A001187 and the first formula. The second program uses the definition of the logarithmic transform, see the Weisstein link and the second formula. The third program uses information about the inverse of the logarithmic transform, see A274804.

			Some a(n) formulas, see A127671:
a(0) = undefined
a(1) = 1*x(1)
a(2) = 1*x(2) - x(1)^2
a(3) = 1*x(3) - 3*x(1)*x(2) + 2*x(1)^3
a(4) = 1*x(4) - 4*x(1)*x(3) - 3*x(2)^2 + 12*x(1)^2*x(2) - 6*x(1)^4
a(5) = 1*x(5) - 5*x(1)*x(4) - 10*x(2)*x(3) + 20*x(1)^2*x(3) + 30*x(1)*x(2)^2 - 60*x(1)^3*x(2) + 24*x(1)^5

Frank Harary and Edgar M. Palmer, Graphical Enumeration, 1973.
Robert James Riddell, Contributions to the theory of condensation, Dissertation, University of Michigan, Ann Arbor, 1951.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 1995, pp. 18-23.

Alois P. Heinz, Table of n, a(n) for n = 1..451
M. Bernstein and N. J. A. Sloane, Some Canonical Sequences of Integers, Linear Algebra and its Applications, Vol. 226-228 (1995), pp. 57-72. Erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms.
Eric W. Weisstein MathWorld, Logarithmic Transform.

Some LOG transform pairs are, n >= 1: A006125(n-1) and A033678(n); A006125(n) and A001187(n); A006125(n+1) and A062740(n); A000045(n) and A112005(n); A000045(n+1) and A007553(n); A000040(n) and A007447(n); A000051(n) and (-1)*A263968(n-1); A002416(n) and A062738(n); A000290(n) and A033464(n-1); A029725(n-1) and A116652(n-1); A052332(n) and A002031(n+1); A027641(n)/A027642(n) and (-1)*A060054(n+1)/(A075180(n-1).
Cf. A274804, A274844, A127671, A000203.
Cf. A112005, A007553, A062740, A007447, A062738, A033464, A116652, A002031, A003704, A003707, A155585, A000142, A226968.

nmax:=22: with(numtheory): b := proc(n): sigma(n) end: a:= proc(n) option remember; b(n) - add(k*binomial(n, k)*b(n-k)*a(k), k=1..n-1)/n: end: seq(a(n), n=1..nmax); # End first LOG program.
nmax:=22: with(numtheory): b := proc(n): sigma(n) end: t1 := log(1 + add(b(n)*x^n/n!, n=1..nmax+1)): t2 := series(t1, x, nmax+1): a := proc(n): n!*coeff(t2, x, n) end: seq(a(n), n=1..nmax); # End second LOG program.
nmax:=22: with(numtheory): b := proc(n): sigma(n) end: f := series(exp(add(r(n)*x^n/n!, n=1..nmax+1)), x, nmax+1): d := proc(n): n!*coeff(f, x, n) end: a(1):=b(1): r(1):= b(1): for n from 2 to nmax+1 do r(n) := solve(d(n)-b(n), r(n)): a(n):=r(n): od: seq(a(n), n=1..nmax); # End third LOG program.

a[1] = 1; a[n_] := a[n] = DivisorSigma[1, n] - Sum[k*Binomial[n, k] * DivisorSigma[1, n-k]*a[k], {k, 1, n-1}]/n; Table[a[n], {n, 1, 22}] (* Jean-François Alcover, Feb 27 2017 *)

N=33; x='x+O('x^N); Vec(serlaplace(log(1+sum(n=1,N,sigma(n)*x^n/n!)))) \\ Joerg Arndt, Feb 27 2017

a(n) = b(n) - Sum_{k = 1..n-1}((k*binomial(n, k)*b(n-k)*a(k))/n), n >= 1, with b(n) = A000203(n) = sigma(n).

E.g.f. log(1+ Sum_{n >= 1}(b(n)*x^n/n!)), n >= 1, with b(n) = A000203(n) = sigma(n).

A346974 Expansion of e.g.f. log( 1 + (exp(x) - 1)^2 / 2 ).

Original entry on oeis.org

1, 3, 4, -15, -134, -357, 2374, 33645, 133186, -1288617, -24887906, -130115895, 1666879306, 40612637523, 262868197414, -4221449488635, -123802488449774, -952293015617937, 18497401668708334, 632675912865355425, 5622243546094977946, -128799294291220310997

Offset: 2

Ilya Gutkovskiy, Aug 09 2021

sign

Cf. A000225, A003704, A060311, A346975, A346976, A346977.

nmax = 23; CoefficientList[Series[Log[1 + (Exp[x] - 1)^2/2], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 2] &
a[n_] := a[n] = StirlingS2[n, 2] - (1/n) Sum[Binomial[n, k] StirlingS2[n - k, 2] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 2, 23}]

a(n) = Stirling2(n,2) - (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling2(n-k,2) * k * a(k).
a(n) ~ -n! * 2^(n+1) * cos(n*arctan(2*arctan(sqrt(2))/log(3))) / (n * (4*arctan(sqrt(2))^2 + log(3)^2)^(n/2)). - Vaclav Kotesovec, Aug 09 2021
a(n) = Sum_{k=1..floor(n/2)} (-1)^(k-1) * (2*k)! * Stirling2(n,2*k)/(k * 2^k). - Seiichi Manyama, Jan 23 2025

A331978 E.g.f.: -log(2 - cosh(x)) (even powers only).

Original entry on oeis.org

0, 1, 4, 46, 1114, 46246, 2933074, 263817646, 31943268634, 5009616448246, 987840438629794, 239217148602642046, 69790939492563608554, 24143849395162438623046, 9772368696995766705116914, 4575221153658910691872135246, 2453303387149157947685779986874

Offset: 0

Ilya Gutkovskiy, Feb 03 2020

nonn

Cf. A000111, A000182, A000629, A003703, A003704, A003707, A094088, A331611.

ptan := proc(n) option remember;
    if irem(n, 2) = 0 then 0 else
    add(`if`(k=0, 1, binomial(n, k)*ptan(n - k)), k = 0..n-1, 2) fi end:
A331978 := n -> ptan(2*n - 1):
seq(A331978(n), n = 0..16);  # Peter Luschny, Jun 06 2022

nmax = 16; Table[(CoefficientList[Series[-Log[2 - Cosh[x]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

a(0) = 0; a(n) = A094088(n) - (1/n) * Sum_{k=1..n-1} binomial(2*n,2*k) * A094088(n-k) * k * a(k).

a(n) ~ (2*n)! / (n * log(2 + sqrt(3))^(2*n)). - Vaclav Kotesovec, Feb 07 2020

A346390 Expansion of e.g.f. -log( 1 - (exp(x) - 1)^3 / 3! ).

Original entry on oeis.org

1, 6, 25, 100, 511, 3626, 30045, 262800, 2470171, 25889446, 302003065, 3821936300, 51672723831, 745789322466, 11505096936085, 189023074558600, 3288243760145491, 60319276499454686, 1164282909466221105, 23603464830964817700, 501435697062735519151

Offset: 3

Ilya Gutkovskiy, Aug 08 2021

nonn

Cf. A000392, A000629, A003704, A327504, A346894, A346954, A346955.

nmax = 23; CoefficientList[Series[-Log[1 - (Exp[x] - 1)^3/3!], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 3] &
a[n_] := a[n] = StirlingS2[n, 3] + (1/n) Sum[Binomial[n, k] StirlingS2[n - k, 3] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 3, 23}]

my(x='x+O('x^25)); Vec(serlaplace(-log(1-(exp(x)-1)^3/3!))) \\ Michel Marcus, Aug 09 2021

a(n) = Stirling2(n,3) + (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling2(n-k,3) * k * a(k).
a(n) ~ (n-1)! / (log(6^(1/3)+1))^n. - Vaclav Kotesovec, Aug 09 2021
a(n) = Sum_{k=1..floor(n/3)} (3*k)! * Stirling2(n,3*k)/(k * 6^k). - Seiichi Manyama, Jan 23 2025

A346954 Expansion of e.g.f. -log( 1 - (exp(x) - 1)^4 / 4! ).

Original entry on oeis.org

1, 10, 65, 350, 1736, 9030, 60355, 561550, 6188996, 69919850, 781211795, 8854058850, 106994019406, 1433756147470, 21287253921635, 339206526695750, 5630710652048216, 96341917117951890, 1708973354556320875, 31787279786739738250, 623964823224788294426

Offset: 4

Ilya Gutkovskiy, Aug 08 2021

nonn

Cf. A000453, A000629, A003704, A327505, A346390, A346895, A346955.

nmax = 24; CoefficientList[Series[-Log[1 - (Exp[x] - 1)^4/4!], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 4] &
a[n_] := a[n] = StirlingS2[n, 4] + (1/n) Sum[Binomial[n, k] StirlingS2[n - k, 4] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 4, 24}]

a(n) = Stirling2(n,4) + (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling2(n-k,4) * k * a(k).
a(n) ~ (n-1)! / (log(2^(3/4)*3^(1/4) + 1))^n. - Vaclav Kotesovec, Aug 09 2021
a(n) = Sum_{k=1..floor(n/4)} (4*k)! * Stirling2(n,4*k)/(k * 24^k). - Seiichi Manyama, Jan 23 2025

cp's OEIS Frontend

A202139 Expansion of e.g.f. log(1/(1-artanh(x))).

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A296435 Expansion of e.g.f. log(1 + arcsinh(x)).

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A354056 Product_{n>=1} 1 / (1 - x^n)^(a(n)/n!) = 1 + sinh(x).

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A012494 Expansion of e.g.f. arctan(sin(x)) (odd powers only).

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A331608 E.g.f.: exp(1 / (1 - sinh(x)) - 1).

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A274805 The logarithmic transform of sigma(n).

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A346974 Expansion of e.g.f. log( 1 + (exp(x) - 1)^2 / 2 ).

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Mathematica

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A331978 E.g.f.: -log(2 - cosh(x)) (even powers only).

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