cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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

Views

Author

Ilya Gutkovskiy, Dec 12 2017

Keywords

Examples

			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! + ...

Links

Crossrefs

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

Programs

  • Maple
    S:= series(ln(1+arcsinh(x)),x,51):
seq(coeff(S,x,j)*j!,j=0..50); # Robert Israel, Dec 12 2017
  • Mathematica
    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]!
  • PARI
    Vecrev(Pol(serlaplace(log(1 + asinh(x + O(x^30)))))) \\ Andrew Howroyd, Dec 12 2017

Formula

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

A296464 Expansion of e.g.f. arcsin(arcsin(x)) (odd powers only).

Original entry on oeis.org

1, 2, 28, 1024, 71632, 8192736, 1392793920, 330041217024, 104069101383936, 42159457593506304, 21346870862961183744, 13213529766600134344704, 9818417126704155249954816, 8625630408510010165396070400, 8844234850947343105068735283200, 10467364426053362392901751845683200
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 13 2017

Keywords

Examples

			arcsin(arcsin(x)) = x/1! + 2*x^3/3! + 28*x^5/5! + 1024*x^7/7! + 71632*x^9/9! + ...

Crossrefs

Cf. A001818, A003712, A012063, A296466.

Programs

  • Mathematica
    nmax = 16; Table[(CoefficientList[Series[ArcSin[ArcSin[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
nmax = 16; Table[(CoefficientList[Series[-I Log[Log[I x + Sqrt[1 - x^2]] + Sqrt[1 + Log[I x + Sqrt[1 - x^2]]^2]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]

Formula

E.g.f.: arcsinh(arcsinh(x)) (odd powers only, absolute values).
E.g.f.: -i*log(log(i*x + sqrt(1 - x^2)) + sqrt(1 + log(i*x + sqrt(1 - x^2))^2)), where i is the imaginary unit (odd powers only).
a(n) ~ sqrt(2) * (2*n)! / (sqrt(Pi*sin(2)*n) * sin(1)^(2*n)). - Vaclav Kotesovec, Dec 13 2017

A296466 Expansion of e.g.f. arcsinh(arcsin(x)) (odd powers only).

Original entry on oeis.org

1, 0, 8, 56, 8000, 342144, 68623488, 8295676416, 2411783847936, 584142614728704, 240810283258527744, 96772676958798741504, 54867909992513301282816, 32661008325245409302937600, 24691868812821871169667072000, 20243305132513358736699378892800, 19829947398943934886214249532620800
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 13 2017

Keywords

Examples

			arcsinh(arcsin(x)) = x/1! + 8*x^5/5! + 56*x^7/7! + 8000*x^9/9! + 342144*x^11/11! + 68623488*x^13/13! + ...

Crossrefs

Cf. A001818, A003722, A012248, A296464.

Programs

  • Mathematica
    nmax = 17; Table[(CoefficientList[Series[ArcSinh[ArcSin[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
nmax = 17; Table[(CoefficientList[Series[Log[Sqrt[1 - Log[I x + Sqrt[1 - x^2]]^2] - I Log[I x + Sqrt[1 - x^2]]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]

Formula

E.g.f.: arcsin(arcsinh(x)) (odd powers only, absolute values).
E.g.f.: log(sqrt(1 - log(i*x + sqrt(1 - x^2))^2) - i*log(i*x + sqrt(1 - x^2))), where i is the imaginary unit (odd powers only).
a(n) ~ 2 * (2*n)! / sqrt(Pi*(4 + Pi^2)*n). - Vaclav Kotesovec, Dec 13 2017

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

Views

Author

Ilya Gutkovskiy, Dec 23 2017

Keywords

Examples

			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! + ...

Links

Crossrefs

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

Programs

  • Maple
    a:=series(arcsin(x*exp(x)),x=0,22): seq(n!*coeff(a,x,n),n=0..21); # Paolo P. Lava, Mar 26 2019
  • Mathematica
    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]!
  • PARI
    first(n) = my(x='x+O('x^n)); Vec(serlaplace(asin(exp(x)*x)), -n) \\ Iain Fox, Dec 23 2017

Formula

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

A353818 Product_{n>=1} (1 + a(n)*x^n/n!) = 1 + arcsin(x).

Original entry on oeis.org

1, 0, 1, -4, 29, -174, 1583, -13168, 144153, -1485330, 20127867, -253341144, 3978820221, -57986205900, 1057400360235, -18016221644544, 370244721585681, -6993826454599146, 162968423791332339, -3490951922268853320, 88052648301403014789, -2075060448716599488276
Offset: 1

Views

Author

Ilya Gutkovskiy, May 08 2022

Keywords

Crossrefs

Cf. A001818, A006973, A137852, A353607, A353819, A353820, A353821.

Programs

  • Mathematica
    nn = 22; f[x_] := Product[(1 + a[n] x^n/n!), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1 - ArcSin[x], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten

A353819 Product_{n>=1} (1 + a(n)*x^n/n!) = 1 + arcsinh(x).

Original entry on oeis.org

1, 0, -1, 4, -11, 66, -547, 4880, -27351, 263310, -3258663, 39791016, -390445563, 5477278548, -84140635815, 1486404086016, -18431412645519, 322018685539542, -6436900596281679, 133183534639917240, -2208721087854287811, 49383164607876494604, -1149793471388581053219
Offset: 1

Views

Author

Ilya Gutkovskiy, May 08 2022

Keywords

Crossrefs

Cf. A001818, A006973, A137852, A353608, A353818, A353820, A353821.

Programs

  • Mathematica
    nn = 23; f[x_] := Product[(1 + a[n] x^n/n!), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1 - ArcSinh[x], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten

A178575 Number of permutations of {1,2,...,3n} whose cycle lengths are all divisible by 3.

Original entry on oeis.org

1, 2, 160, 62720, 68992000, 163235072000, 710399033344000, 5129081020743680000, 57096929922918645760000, 927825111247427993600000000, 21095031729321522862489600000000, 648714415740095471067280179200000000, 26246985260844262759382156050432000000000
Offset: 0

Views

Author

Geoffrey Critzer, Dec 23 2010

Keywords

Examples

			a(1) = 2 because we have (123) and (132).

References

  • Herbert S. Wilf, Generatingfunctiontology, page 209

Links

Crossrefs

Cf. A001818, A258878.

Programs

  • Maple
    a:= n-> factorial(3*n)*(mul(1+3*i, i = 1 .. n-1))/(factorial(n)*3^n): seq(a(n), n = 0 .. 11);
  • Mathematica
    Table[(-1)^(n/3) Binomial[-1/3,n/3]n!,{n,0,30,3}]
  • PARI
    v=Vec(serlaplace(1/(1-x^3+O(x^50))^(1/3))); vector(#v\3,n,v[3*n-2])

Formula

a(n) = (-1)^(n/3)*binomial(-1/3,n/3)*n!.
E.g.f.: 1/(1-x^3)^(1/3).
a(n) = ((3*n)!/(n!*3^n))*Product_{i=1..n-1} (1+3*i) (from the Wilf reference).
a(n) ~ (3*n)! / (Gamma(1/3) * n^(2/3)). - Vaclav Kotesovec, Jun 15 2015
D-finite with recurrence: a(n) = (3*n-1)*(3*n-2)^2*a(n-1), a(0)=1. - Georg Fischer, Jul 02 2021 (from the 3rd formula)

A291560 E.g.f. A(x,k) satisfies: sin(A(x,k)) = k * sin(x).

Original entry on oeis.org

1, -1, 1, 1, -10, 9, -1, 91, -315, 225, 1, -820, 8694, -18900, 11025, -1, 7381, -224730, 1143450, -1819125, 893025, 1, -66430, 5684679, -61647300, 203378175, -255405150, 108056025, -1, 597871, -142714845, 3162834675, -19494349875, 47377655325, -49165491375, 18261468225, 1, -5380840, 3573251964, -158546770200, 1734021238950, -7311738634200, 14041664336700, -12417798393000, 4108830350625, -1, 48427561, -89379726660, 7858123038900, -148224512094750, 1025176095093150, -3257761647640500, 5167045911327300, -3981456609755625, 1187451971330625
Offset: 1

Views

Author

Paul D. Hanna, Aug 26 2017

Keywords

Comments

Compare to the law of sines of a spherical triangle: sin(A)/sin(a) = k.
The series reversion of e.g.f. A(x,k) wrt x equals A(x, 1/k).

Examples

			This triangle of coefficients T(n,r) in e.g.f. A(x,k) begins:
[1],
[-1, 1],
[1, -10, 9],
[-1, 91, -315, 225],
[1, -820, 8694, -18900, 11025],
[-1, 7381, -224730, 1143450, -1819125, 893025],
[1, -66430, 5684679, -61647300, 203378175, -255405150, 108056025],
[-1, 597871, -142714845, 3162834675, -19494349875, 47377655325, -49165491375, 18261468225],
[1, -5380840, 3573251964, -158546770200, 1734021238950, -7311738634200, 14041664336700, -12417798393000, 4108830350625],
[-1, 48427561, -89379726660, 7858123038900, -148224512094750, 1025176095093150, -3257761647640500, 5167045911327300, -3981456609755625, 1187451971330625],
[1, -435848050, 2234929014549, -387282522072600, 12391233508580850, -136052492985945900, 674608025957515650, -1713147048499887000, 2313226290268018125, -1579311121869731250, 428670161650355625], ...
where e.g.f. A(x,k) = Sum_{n>=1, r=1..n} T(n,r) * x^(2*n-1) * k^(2*r-1) / (2*n-1)!.
E.g.f.: A(x,k) = k*x + (k^3 - k)*x^3/3! + (9*k^5 - 10*k^3 + k)*x^5/5! + (225*k^7 - 315*k^5 + 91*k^3 - k)*x^7/7! + (11025*k^9 - 18900*k^7 + 8694*k^5 - 820*k^3 + k)*x^9/9! + (893025*k^11 - 1819125*k^9 + 1143450*k^7 - 224730*k^5 + 7381*k^3 - k)*x^11/11! + (108056025*k^13 - 255405150*k^11 + 203378175*k^9 - 61647300*k^7 + 5684679*k^5 - 66430*k^3 + k)*x^13/13! + (18261468225*k^15 - 49165491375*k^13 + 47377655325*k^11 - 19494349875*k^9 + 3162834675*k^7 - 142714845*k^5 + 597871*k^3 - k)*x^15/15! + (4108830350625*k^17 - 12417798393000*k^15 + 14041664336700*k^13 - 7311738634200*k^11 + 1734021238950*k^9 - 158546770200*k^7 + 3573251964*k^5 - 5380840*k^3 + k)*x^17/17! + (1187451971330625*k^19 - 3981456609755625*k^17 + 5167045911327300*k^15 - 3257761647640500*k^13 + 1025176095093150*k^11 - 148224512094750*k^9 + 7858123038900*k^7 - 89379726660*k^5 + 48427561*k^3 - k)*x^19/19! +...
such that sin(A(x,k)) = k * sin(x).

Links

Crossrefs

Cf. A002452 (column 1), A001818 (diagonal), A291561 (diagonal), A291562 (central terms).
Cf. A291527 (variant).

Programs

  • Mathematica
    T[n_, k_] := If[ n < 1, 0, (2 n - 1)! Coefficient[ SeriesCoefficient[ ArcSin[y Sin[x]], {x, 0, 2 n - 1}], y, 2 k - 1]]; (* Michael Somos, Jul 03 2018 *)
T[n_, k_] := ((-1)^n/((2*k - 1)^2*4^(2*k - 1)))*((2*k)!/k!)^2 * Sum[((-1)^i*(2*i - 1)^(2*n - 1))/((k - i)!*(k + i - 1)!), {i, 1, n}]; (* Vjekoslav-Leonard Prcic, Oct 10 2018 *)
  • PARI
    {T(n, r) = (2*n-1)! * polcoeff( polcoeff( asin( k*sin(x + O(x^(2*n)))), 2*n-1,x), 2*r-1, k)}
for(n=1, 10, for(r=1, n, print1(T(n, r), ", ")); print(""))

Formula

E.g.f. A(x,k) = Sum_{n>=1, r=1..n} T(n,r) * x^(2*n-1) * k^(2*r-1)/(2*n-1)!, satisfies:
(1) sin(A(x,k)) = k * sin(x).
(2) A(x,k) = asin(k * sin(x)).
(3) A( A(x,k), 1/k) = x.
(4) sin( A^r(x,k) ) = k^r * sin(x) where A^r(x,k) = A(x,k^r) is the r-th iteration of A(x,k) wrt x, with A^0(x,k) = x.
(5) A(x,1) = x.
Row sums of n-th row equals zero for n>1.
T(n+1,1) = (-1)^n for n>=0.
T(n+1,2) = (-1)^(n-1) * (9^n - 1)/8 for n>=1.
T(n+1,n+1) = ( (2*n)! / (n!*2^n) )^2 = A001818(n) for n>=0.
T(n, r) = (-1)^n / ((2*r - 1)^2 * 4^(2*r - 1)) * ((2*r)! / r!)^2 * Sum_{i=1..n} (-1)^i * (2*i - 1)^(2*n - 1) / ((r - i)! * (r + i - 1)!). - Vjekoslav-Leonard Prcic, Oct 10 2018

A296679 Expansion of e.g.f. arcsinh(arctanh(x)) (odd powers only).

Original entry on oeis.org

1, 1, 13, 341, 18649, 1599849, 205524837, 36391450941, 8546308276401, 2564025898856913, 957697868873929149, 435619128300038521893, 237104370189582892175241, 152148421079949399306125625, 113672892845152570858515803925, 97820056722556900357454981990925
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 18 2017

Keywords

Examples

			arcsinh(arctanh(x)) = x/1! + x^3/3! + 13*x^5/5! + 341*x^7/7! + 18649*x^9/9! + 1599849*x^11/11! + ...

Crossrefs

Cf. A001818, A003706, A010050, A012254, A096717, A096718, A296464, A296466, A296677.

Programs

  • Mathematica
    nmax = 16; Table[(CoefficientList[Series[ArcSinh[ArcTanh[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
nmax = 16; Table[(CoefficientList[Series[Log[(Log[1 + x] - Log[1 - x])/2 + Sqrt[1 + (Log[1 + x] - Log[1 - x])^2/4]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]

Formula

E.g.f.: arcsin(arctan(x)) (odd powers only, absolute values).
E.g.f.: log((log(1 + x) - log(1 - x))/2 + sqrt(1 + (log(1 + x) - log(1 - x))^2/4)) (odd powers only).

A296680 Expansion of e.g.f. arcsin(arctanh(x)) (odd powers only).

Original entry on oeis.org

1, 3, 53, 2359, 198953, 27412011, 5625656541, 1613676694239, 617477049181521, 304167421243513683, 187546541676182230149, 141512355477854459198343, 128265950128144233675269241, 137512081213377707268891639675, 172108297920263623816775456321325
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 18 2017

Keywords

Examples

			arcsin(arctanh(x)) = x/1! + 3*x^3/3! + 53*x^5/5! + 2359*x^7/7! + 198953*x^9/9! + 27412011*x^11/11! + ...

Links

Crossrefs

Cf. A001818, A003717, A010050, A012134, A296464, A296466, A296679.

Programs

  • Maple
    S:= series(arcsin(arctanh(x)),x,52):
seq(coeff(S,x,n)*n!,n=1..51,2); # Robert Israel, Dec 18 2017
  • Mathematica
    nmax = 15; Table[(CoefficientList[Series[ArcSin[ArcTanh[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
nmax = 15; Table[(CoefficientList[Series[-I Log[(I/2) (Log[1 + x] - Log[1 - x]) + Sqrt[1 - (Log[1 + x] - Log[1 - x])^2/4]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]

Formula

E.g.f.: arcsinh(arctan(x)) (odd powers only, absolute values).
E.g.f.: -i*log((i/2)*(log(1 + x) - log(1 - x)) + sqrt(1 - (log(1 + x) - log(1 - x))^2/4)), where i is the imaginary unit (odd powers only).
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