A216401 -id:A216401 - cp's OEIS frontend

A009768 Expansion of e.g.f. tanh(exp(x)*x).

0, 1, 2, 1, -20, -159, -594, 2465, 69560, 665665, 1593850, -67177791, -1413216540, -12990964063, 64480265318, 4811655319393, 90259507840240, 540272971703937, -20890652777843598, -798235260367432831, -12766815370452348580

Offset: 0

R. H. Hardin

Seiichi Manyama, Table of n, a(n) for n = 0..100

Cf. A009121, A009565, A009635, A216401.

With[{nn=20},CoefficientList[Series[Tanh[Exp[x]*x],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 03 2023 *)

my(x='x+O('x^30)); concat(0, Vec(serlaplace(tanh(exp(x)*x)))) \\ Michel Marcus, Oct 01 2021

Extended with signs by Olivier Gérard, Mar 15 1997

Previous Mathematica program replaced by Harvey P. Dale, Jun 03 2023

A191719 Expansion of e.g.f. arctan(x*exp(x)).

Original entry on oeis.org

0, 1, 2, 1, -20, -151, -354, 6217, 100472, 537777, -7631270, -223395919, -2120164188, 22050300505, 1154262915638, 17130776734905, -105423782758544, -11372993234072863, -245877012220234446, 345837436238423521, 188329590656514108380

Offset: 0

Vladimir Kruchinin, Jun 13 2011

sign

Seiichi Manyama, Table of n, a(n) for n = 0..100

Cf. A009635, A216401, A297009.

Rest[CoefficientList[Series[ArcTan[x*Exp[x]],{x,0,20}],x]*Range[0,20]!] (* Vaclav Kotesovec, Jan 02 2014 *)

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

a(n) = n!*Sum_{m=1..(n+1)/2} ((2*m-1)^(n-2*m)*(-1)^(m-1))/(n-2*m+1)!.

a(n) ~ (n-1)! * sin(n*arctan(1/tan(r))) * (cos(r)/r)^n, where r = Im(LambertW(I)) = A305200 = 0.576412723031435283148289239887... is the root of the equation exp(r*tan(r))=cos(r)/r. - Vaclav Kotesovec, Jan 02 2014

a(0)=0 prepended by Seiichi Manyama, Oct 01 2021

A297009 Expansion of e.g.f. arcsin(x*exp(x)).

Original entry on oeis.org

0, 1, 2, 4, 16, 104, 816, 7792, 89216, 1177920, 17603200, 294334976, 5442281472, 110221745152, 2426850793472, 57718658411520, 1474590580228096, 40274407232294912, 1171043235561185280, 36115912820342407168, 1177554628069200035840, 40471207964013864124416

Offset: 0

Ilya Gutkovskiy, Dec 23 2017

nonn

			arcsin(x*exp(x)) = x^1/1! + 2*x^2/2! + 4*x^3/3! + 16*x^4/4! + 104*x^5/5! + 816*x^6/6! + ...

Seiichi Manyama, Table of n, a(n) for n = 0..100

Cf. A001818, A009017, A009121, A009448, A009565, A009635, A009768, A191719, A216401, A297010.

a:=series(arcsin(x*exp(x)),x=0,22): seq(n!*coeff(a,x,n),n=0..21); # Paolo P. Lava, Mar 26 2019

nmax = 21; CoefficientList[Series[ArcSin[x Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[-I Log[I x Exp[x] + Sqrt[1 - x^2 Exp[2 x]]], {x, 0, nmax}], x] Range[0, nmax]!

first(n) = my(x='x+O('x^n)); Vec(serlaplace(asin(exp(x)*x)), -n) \\ Iain Fox, Dec 23 2017

a(n) ~ sqrt(1 + LambertW(1)) * n^(n-1) / (exp(n) * LambertW(1)^n). - Vaclav Kotesovec, Mar 26 2019

A297010 Expansion of e.g.f. arcsinh(x*exp(x)).

Original entry on oeis.org

0, 1, 2, 2, -8, -76, -264, 1672, 36800, 261648, -1443680, -66164704, -792152448, 2482671424, 289529373056, 5294082629760, 1648955815936, -2474170098704128, -65494141255724544, -303927676523118080, 35926135133071923200, 1341060635191667045376

Offset: 0

Ilya Gutkovskiy, Dec 23 2017

sign

			arcsinh(x*exp(x)) = x^1/1! + 2*x^2/2! + 2*x^3/3! - 8*x^4/4! - 76*x^5/5! - 264*x^6/6! + ...

Seiichi Manyama, Table of n, a(n) for n = 0..100

Cf. A001818, A009017, A009121, A009448, A009565, A009635, A009768, A191719, A216401, A297009.

a:=series(arcsinh(x*exp(x)),x=0,22): seq(n!*coeff(a,x,n),n=0..21); # Paolo P. Lava, Mar 26 2019

nmax = 21; CoefficientList[Series[ArcSinh[x Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Log[x Exp[x] + Sqrt[1 + x^2 Exp[2 x]]], {x, 0, nmax}], x] Range[0, nmax]!

first(n) = my(x='x+O('x^n)); Vec(serlaplace(asinh(exp(x)*x)), -n) \\ Iain Fox, Dec 23 2017

A275385 Number of labeled functional digraphs on n nodes with only odd sized cycles and such that every vertex is at a distance of at most 1 from a cycle.

Original entry on oeis.org

1, 1, 3, 12, 73, 580, 5601, 63994, 844929, 12647016, 211616065, 3914510446, 79320037281, 1747219469164, 41569414869633, 1062343684252530, 29023112392093441, 844101839207139280, 26038508978625589377, 849150487829425227094, 29189561873274715264545

Offset: 0

Geoffrey Critzer, Jul 25 2016

nonn

Equivalently, these are the functions counted by A116956 with the additional constraint that every element is mapped to a recurrent element. A recurrent element is an element on a cycle in the functional digraph.

Alois P. Heinz, Table of n, a(n) for n = 0..411

Cf. A000246, A006153, A009444, A116956, A216401.

b:= proc(n) option remember; `if`(n=0, 1, add(`if`(j::odd,
       (j-1)!*b(n-j)*binomial(n-1, j-1), 0), j=1..n))
    end:
a:= n-> add(b(j)*j^(n-j)*binomial(n, j), j=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 25 2016

nn = 20; Range[0, nn]! CoefficientList[Series[Sqrt[(1 + z*Exp[z])/(1 - z*Exp[z])], {z, 0, nn}], z]

default(seriesprecision, 30);
S=sqrt((1 + x*exp(x))/(1 - x*exp(x)));
v=Vec(S); for(n=2,#v-1,v[n+1]*=n!); v \\ Charles R Greathouse IV, Jul 29 2016

E.g.f.: sqrt((1 + z*exp(z))/(1 - z*exp(z))).
Exponential transform of A216401.
a(n) ~ 2 * n^n / (sqrt(1+LambertW(1)) * LambertW(1)^n * exp(n)). - Vaclav Kotesovec, Jun 26 2022

A294312 Expansion of e.g.f. sec(x*exp(x)).

Original entry on oeis.org

1, 0, 1, 6, 29, 180, 1501, 14434, 154265, 1856232, 24953401, 368767102, 5936244533, 103519338780, 1944554725205, 39134556793050, 840024295910833, 19157944025344464, 462629389438242673, 11792248121970820598, 316398168231432879565, 8913743651504295251844

Offset: 0

Ilya Gutkovskiy, Dec 27 2017

nonn

			sec(x*exp(x)) = 1 + x^2/2! + 6*x^3/3! + 29*x^4/4! + 180*x^5/5! + 1501*x^6/6! + ...

Cf. A000364, A009007, A009017, A009121, A009300, A009448, A009565, A009635, A009768, A139134, A191719, A216401, A217502, A294313, A297009, A297010.

a:=series(sec(x*exp(x)),x=0,22): seq(n!*coeff(a,x,n),n=0..21); # Paolo P. Lava, Mar 27 2019

nmax = 21; CoefficientList[Series[Sec[x Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[1/Cos[x Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!

A294313 Expansion of e.g.f. sech(x*exp(x)).

Original entry on oeis.org

1, 0, -1, -6, -19, 20, 899, 7966, 27705, -366552, -8374201, -80690302, 9794597, 16015845820, 317370642315, 2554368906150, -37571987331343, -1784464543440304, -31315944840101233, -80221319702865398, 12685422355781995485, 422083364962616527716

Offset: 0

Ilya Gutkovskiy, Dec 27 2017

sign

			sech(x*exp(x)) = 1 - x^2/2! - 6*x^3/3! - 19*x^4/4! + 20*x^5/5! + 899*x^6/6! + ...

Cf. A000364, A009017, A009121, A009301, A009448, A009565, A009635, A009768, A191719, A216401, A217502, A294312, A296544, A297009, A297010.

a:=series(sech(x*exp(x)),x=0,22): seq(n!*coeff(a,x,n),n=0..21); # Paolo P. Lava, Mar 27 2019

nmax = 21; CoefficientList[Series[Sech[x Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[1/Cosh[x Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!

cp's OEIS Frontend

A009768 Expansion of e.g.f. tanh(exp(x)*x).

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A191719 Expansion of e.g.f. arctan(x*exp(x)).

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Mathematica

Maxima

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A297009 Expansion of e.g.f. arcsin(x*exp(x)).

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A297010 Expansion of e.g.f. arcsinh(x*exp(x)).

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A275385 Number of labeled functional digraphs on n nodes with only odd sized cycles and such that every vertex is at a distance of at most 1 from a cycle.

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Maple

Mathematica

PARI

Formula

A294312 Expansion of e.g.f. sec(x*exp(x)).

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Maple

Mathematica

A294313 Expansion of e.g.f. sech(x*exp(x)).

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Maple

Mathematica