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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A274703 Exponential generating function 1/M_{3}(z^3) where M_{n}(z) is the n-th Mittag-Leffler function, nonzero coefficients only.

Original entry on oeis.org

1, -4, 133, -15130, 4101799, -2177360656, 1999963458217, -2919514870785766, 6365117686550339275, -19765974970578036695068, 84220118333781814726917709, -477722110504065444764182065202, 3518554409906597166261453268226671, -32952557456293494405944914420304822440
Offset: 0

Views

Author

Peter Luschny, Jul 03 2016

Keywords

Comments

For references see also A274705 which is the main entry for this sequence of sequences.

Links

Crossrefs

Cf. A181983 (n=1), A009843 (n=2), A274704 (n=4), A274705 (array).

Programs

  • Maple
    s := series(z/((exp(z)+2*exp(-z/2)*cos(z*3^(1/2)/2))/3),z,60):
seq((n*3+1)!*coeff(s,z,n*3+1), n=0..13);
  • Mathematica
    c = CoefficientList[Series[1/MittagLefflerE[3,z^3],{z,0,15*3}],z];
Table[Factorial[3*n+1]*c[[3*n+1]], {n,0,13}]

Formula

E.g.f. (nonzero coefficients): z/((exp(z)+2*exp(-z/2)*cos(z*3^(1/2)/2))/3).
For n >= 1, a(n) = -Sum_{k=0..n-1} a(k) binomial(3n+1,3k+1). - Robert Israel, Jul 03 2016

A274704 Exponential generating function 1/M_{4}(z^4) where M_{n}(z) is the n-th Mittag-Leffler function, nonzero coefficients only.

Original entry on oeis.org

1, -5, 621, -437593, 1026405753, -6054175060941, 75477454065058725, -1766732850877953050849, 71248914440011028226682737, -4637564239713542128355021380117, 462852368857623061805761137170608989, -67965094887205237792816627191801312013545
Offset: 0

Views

Author

Peter Luschny, Jul 03 2016

Keywords

Links

Crossrefs

Cf. A181983 (n=1), A009843 (n=2), A274703 (n=3), A274705 (array).

Programs

  • Maple
    s := series(2*z/(cosh(z)+cos(z)),z,60):
seq((4*n+1)!*coeff(s,z,4*n+1),n=0..11);
  • Mathematica
    c = CoefficientList[Series[1/MittagLefflerE[4, z^4], {z, 0, 15*4}], z];
Table[Factorial[4*n+1]*c[[4*n+1]], {n, 0, 12}]

Formula

E.g.f. (nonzero coefficients): 2*z/(cosh(z)+cos(z)).
For n >= 1, a(n) = - Sum_{k=0..n-1} a(k)*binomial(4*k+1,4*n+1). - Robert Israel, Jul 04 2016

A294314 Expansion of e.g.f. log(1 + x*sec(x))*exp(x).

Original entry on oeis.org

0, 1, 1, 5, 0, 64, -245, 2757, -23576, 272256, -3270977, 45055845, -671589952, 10984688636, -193875825117, 3688182769117, -75085512079184, 1630385857436224, -37596306847103457, 917765946045581357, -23641953753495247624, 640958728426947233468, -18242640219843554954221
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 27 2017

Keywords

Examples

			log(1 + x*sec(x))*exp(x) = x/1! + x^2/2! + 5*x^3/3! + 64*x^5/5! - 245*x^6/6! + ...

Crossrefs

Cf. A009334, A009371, A009442, A009843, A295278, A296336, A296436, A296438, A297206.

Programs

  • Maple
    a:=series(log(1+x*sec(x))*exp(x),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 27 2019
  • Mathematica
    nmax = 22; CoefficientList[Series[Log[1 + x Sec[x]] Exp[x], {x, 0, nmax}], x] Range[0, nmax]!

A295278 Expansion of e.g.f. log(1 + x*sech(x))*exp(x).

Original entry on oeis.org

0, 1, 1, -1, 0, 4, -5, 13, -392, 2112, 7663, -165067, 1011560, -2965756, -11164309, 630876517, -12760548400, 133046910432, -189966787521, -18567623055795, 392188656574896, -5061972266268844, 33655544331988203, 565132153437469165, -26647451471277927416
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 27 2017

Keywords

Examples

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

Crossrefs

Cf. A009353, A009390, A009443, A009843, A294314, A296336, A296437, A296439, A297206.

Programs

  • Maple
    a:=series(log(1+x*sech(x))*exp(x),x=0,25): seq(n!*coeff(a,x,n),n=0..24); # Paolo P. Lava, Mar 27 2019
  • Mathematica
    nmax = 24; CoefficientList[Series[Log[1 + x Sech[x]] Exp[x], {x, 0, nmax}], x] Range[0, nmax]!

A296741 Expansion of e.g.f. arcsin(x*sec(x)) (odd powers only).

Original entry on oeis.org

1, 4, 64, 2752, 237312, 34390016, 7512117248, 2302977392640, 942529341030400, 496287845973753856, 326775812392982937600, 263039306566659448242176, 254121613033387345942937600, 290175686081926976733941071872, 386599796043915196967089006968832
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 19 2017

Keywords

Examples

			arcsin(x*sec(x)) = x/1! + 4*x^3/3! + 64*x^5/5! + 2752*x^7/7! + 237312*x^9/9! + ...

Crossrefs

Cf. A001818, A003700, A009118, A009119, A009562, A009563, A009765, A009843, A102072, A191003, A296464, A296466, A296679, A296680, A296742, A296743.

Programs

  • Mathematica
    nmax = 15; Table[(CoefficientList[Series[ArcSin[x Sec[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
  • PARI
    first(n) = x='x+O('x^(2*n)); vecextract(Vec(serlaplace(asin(x/cos(x)))), (4^n - 1)/3) \\ Iain Fox, Dec 19 2017

Formula

a(n) = (2*n+1)! * [x^(2*n+1)] arcsin(x*sec(x)).

A296742 Expansion of e.g.f. arcsinh(x*sec(x)) (odd powers only).

Original entry on oeis.org

1, 2, 4, -8, 2448, 130976, -2342848, -239130240, 99052990720, 8918588764672, -2795242017684480, -92786315822417920, 279479081010906828800, -57316070780459900928, -39411396653183724314673152, 5932051008707372732672475136, 10689040617354387626585873252352
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 19 2017

Keywords

Examples

			arcsinh(x*sec(x)) = x/1! + 2*x^3/3! + 4*x^5/5! - 8*x^7/7! + 2448*x^9/9! + ...

Crossrefs

Cf. A001818, A003700, A009118, A009119, A009562, A009563, A009765, A009843, A102072, A191003, A296464, A296466, A296679, A296680, A296741, A296743.

Programs

  • Mathematica
    nmax = 17; Table[(CoefficientList[Series[ArcSinh[x Sec[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
  • PARI
    first(n) = x='x+O('x^(2*n)); vecextract(Vec(serlaplace(asinh(x/cos(x)))), (4^n - 1)/3) \\ Iain Fox, Dec 19 2017

Formula

a(n) = (2*n+1)! * [x^(2*n+1)] arcsinh(x*sec(x)).

A296743 Expansion of e.g.f. arctanh(x*sec(x)) (odd powers only).

Original entry on oeis.org

1, 5, 109, 5977, 612729, 100954061, 24395453861, 8128143367905, 3571195811862385, 2000535014776893973, 1391684597704875555165, 1177047158822263838854889, 1189444022487013498606939625, 1415364934488337503351305867997, 1958850511524588636608881908473749
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 19 2017

Keywords

Examples

			arctanh(x*sec(x)) = x/1! + 5*x^3/3! + 109*x^5/5! + 5977*x^7/7! + 612729*x^9/9! + ...

Crossrefs

Cf. A003700, A009118, A009119, A009562, A009563, A009765, A009843, A010050, A102075, A191003, A296465, A296467, A296741, A296742.

Programs

  • Mathematica
    nmax = 15; Table[(CoefficientList[Series[ArcTanh[x Sec[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
  • PARI
    first(n) = x='x+O('x^(2*n)); vecextract(Vec(serlaplace(atanh(x/cos(x)))), (4^n - 1)/3) \\ Iain Fox, Dec 19 2017

Formula

a(n) = (2*n+1)! * [x^(2*n+1)] arctanh(x*sec(x)).

A009391 Expansion of log(1 + tanh(x))/cos(x).

Original entry on oeis.org

0, 1, -1, 3, -4, 25, -61, 427, -1184, 12465, -49201, 555731, -2361844, 35135945, -191422141, 2990414715, -17147588384, 329655706465, -2289437638081, 45692713833379, -329955144475204, 7777794952988025, -65643617893832221
Offset: 0

Views

Author

R. H. Hardin

Keywords

Crossrefs

a(2n+1) = A009843(n), |a(2n)| = A009391(2n).

Programs

  • Mathematica
    CoefficientList[Series[Log[1 + Tanh[x]]*Sec[x], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 24 2015 *)

Formula

a(n) ~ n! * 2^(n+1) * (Pi + (1+(-1)^n) * log(2/(1+exp(Pi)))) / Pi^(n+1). - Vaclav Kotesovec, Jan 24 2015

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997
Incorrect Mathematica program deleted by Harvey P. Dale, Oct 14 2022

A193472 Numerator of ez(n-1)*n!/(4^n-2^n) where ez(n) is the n-th coefficient of sec(t)+tan(t) for n>0, a(0) = 1.

Original entry on oeis.org

1, 1, 1, 3, 1, 25, 1, 427, 1, 12465, 5, 555731, 691, 35135945, 7, 2990414715, 3617, 329655706465, 43867, 45692713833379, 174611, 1111113564712575, 854513, 1595024111042171723, 236364091, 387863354088927172625, 8553103, 110350957750914345093747, 23749461029
Offset: 0

Views

Author

Peter Luschny, Aug 07 2011

Keywords

Links

Crossrefs

Cf. A000367, A002445, A009843, A193475, A193473.

Programs

  • Maple
    gf := (f,n) -> coeff(series(f(x),x,n+1),x,n):
BG := n ->`if`(n=0,1,gf(sec+tan,n-1)*n!/(4^n-2^n)):
A193472 := n -> numer(BG(n)): seq(A193472(n),n=0..28);
  • Mathematica
    ez[n_] := SeriesCoefficient[Sec[t] + Tan[t], {t, 0, n}];
a[0] = 1; a[n_] := Numerator[ez[n-1] n!/(4^n - 2^n)];
Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Jun 24 2019 *)

A193473 Denominator of ez(n-1)*n!/(4^n-2^n) where ez(n) is the n-th coefficient of sec(t)+tan(t) for n>0, a(0) = 1.

Original entry on oeis.org

1, 2, 6, 56, 30, 992, 42, 16256, 30, 261632, 66, 4192256, 2730, 67100672, 6, 1073709056, 510, 17179738112, 798, 274877382656, 330, 628292059136, 138, 70368735789056, 2730, 1125899873288192, 6, 18014398375264256, 870
Offset: 0

Views

Author

Peter Luschny, Aug 07 2011

Keywords

Links

Crossrefs

Cf. A000367, A002445, A009843, A193475, A193472.

Programs

  • Maple
    gf := (f,n) -> coeff(series(f(x),x,n+1),x,n):
BG := n ->`if`(n=0,1,gf(sec+tan,n-1)*n!/(4^n-2^n)):
A193473 := n -> denom(BG(n)): seq(A193473(n),n=0..28);
  • Mathematica
    ez[n_] := SeriesCoefficient[Sec[t] + Tan[t], {t, 0, n}];
a[0] = 1; a[n_] := Denominator[ez[n - 1] n!/(4^n - 2^n)];
Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Jun 26 2019 *)
Previous Showing 11-20 of 26 results. Next

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