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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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

A353820 Product_{n>=1} (1 + a(n)*x^n/n!) = 1 + arctan(x).

Original entry on oeis.org

1, 0, -2, 8, -16, 96, -832, 9344, -27648, 238080, -4228608, 55812096, -398991360, 4930609152, -98606039040, 2440552022016, -17762113880064, 235149341884416, -7331825098948608, 170578782435409920, -2009778629489197056, 38563016760590598144, -1278044473427380666368
Offset: 1

Views

Author

Ilya Gutkovskiy, May 08 2022

Keywords

Crossrefs

Cf. A006973, A010050, A137852, A353611, A353818, A353819, A353821.

Programs

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

A353821 Product_{n>=1} (1 + a(n)*x^n/n!) = 1 + arctanh(x).

Original entry on oeis.org

1, 0, 2, -8, 64, -384, 3968, -34432, 414720, -4454400, 68247552, -912236544, 15949529088, -245572583424, 5012834549760, -92436465352704, 2119956936523776, -42836227522560000, 1123874181449515008, -26161653829651660800, 730049769522063212544, -18719979459270521389056
Offset: 1

Views

Author

Ilya Gutkovskiy, May 08 2022

Keywords

Crossrefs

Cf. A006973, A010050, A137852, A353779, A353818, A353819, A353820.

Programs

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

A353610 Product_{n>=1} (1 + a(n)*x^(2*n)/(2*n)!) = sec(x).

Original entry on oeis.org

1, 5, -14, 1777, -14744, 247994, -74928944, 42293543177, -1163849271296, 95795966018440, -44942000161435904, 4494117864138588514, -3539995034294896016384, 770158600620174924566672, -510461123036204706738612224, 1162153458061287151457003978297
Offset: 1

Views

Author

Ilya Gutkovskiy, May 07 2022

Keywords

Crossrefs

Cf. A000364, A006973, A137852, A353607, A353608, A353609, A353611.

Programs

  • Mathematica
    nn = 16; f[x_] := Product[(1 + a[n] x^(2 n)/(2 n)!), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - Sec[x], {x, 0, 2 nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten

A353779 Product_{n>=1} (1 + a(n)*x^n/n!) = 1 + tanh(x).

Original entry on oeis.org

1, 0, -2, 8, -24, 144, -720, 7552, -35840, 427520, -3628800, 45415424, -479001600, 7094226944, -82614884352, 1741160087552, -20922789888000, 371094631612416, -6402373705728000, 137529198176370688, -2379913632645120000, 55730621780175355904
Offset: 1

Views

Author

Ilya Gutkovskiy, May 08 2022

Keywords

Crossrefs

Cf. A006973, A137852, A155585, A353607, A353608, A353609, A353610, A353611.

Programs

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

A353822 Product_{n>=1} (1 + x^n/n!)^a(n) = exp(-x)/(1 - x).

Original entry on oeis.org

0, 1, 2, 9, 24, 110, 720, 5985, 39200, 343224, 3628800, 41295870, 479001600, 6130959120, 87104969952, 1318070979225, 20922789888000, 354344089779680, 6402373705728000, 121882240625961816, 2432849766865689600, 51041049953430700800, 1124000727777607680000
Offset: 1

Views

Author

Ilya Gutkovskiy, May 08 2022

Keywords

Crossrefs

Cf. A000166, A006973, A137852.

Programs

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

A348207 Product_{n>=1} (1 + a(n)*x^n/(n!)^2) = BesselI(0,2*sqrt(x)).

Original entry on oeis.org

1, 1, -8, 129, -2424, 87040, -3354000, 234927105, -13619579120, 1467819193176, -142339701178080, 21415200007555200, -2958022926285910560, 605932431017659471440, -110644439905256239190208, 32132110188849291391675905, -7427852296898683736690604000
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 06 2021

Keywords

Crossrefs

Cf. A006973, A137852, A157159, A157164, A348205, A348206.

Formula

Product_{n>=1} (1 + a(n)*x^n/(n!)^2) = Sum_{n>=0} x^n/(n!)^2.

A159310 G.f.: Product_{n>=1} (1 + a(n)*x^n/n!) = Sum_{n>=0} (n+1)^(n-1)*x^n/n! = LambertW(-x)/(-x).

Original entry on oeis.org

1, 3, 7, 97, 601, 7576, 116929, 2482537, 42814321, 1040362966, 25933795801, 760154969850, 23297606120881, 816970034324900, 29137514248718373, 1194044411689941241, 48661170952876980481, 2227962859999303395766
Offset: 1

Views

Author

Paul D. Hanna, Apr 15 2009

Keywords

Examples

			G.f.: W(x) = (1+x)*(1+3*x^2/2!)*(1+7*x^3/3!)*(1+97*x^4/4!)*(1+601*x^5/5!)* ...
W(x) = 1 + x + 3*x^2/2! + 4^2*x^3/3! + 5^3*x^4/4! + 6^4*x^5/5! + ...
where W(x/exp(x)) = exp(x) and exp(x*W(x)) = W(x) = LambertW(-x)/(-x).

Crossrefs

Cf. A137852.

Programs

  • PARI
    {a(n)=if(n<1, 0, polcoeff(sum(k=0,n,(k+1)^(k-1)*x^k/k!)/prod(k=1, n-1, 1+a(k)*x^k +x*O(x^n)), n))}
  • PARI
    {a(n)=if(n<1, 0, if(n==1, 1,n^(n-1) + (n-1)!*((-1)^n + sumdiv(n, d, if(d1, d*(-a(d)/d!)^(n/d))))))} \\ Paul D. Hanna, Apr 15 2009

Formula

G.f.: Sum_{n>=1} log(1 + a(n)*x^n/n!) = Sum_{n>=1} n^(n-1)*x^n/n! = -LambertW(-x).
G.f.: Sum_{n>=1} log(1 + a(n)*exp(-n*x)*x^n/n!) = x.
From Paul D. Hanna, Apr 15 2009: (Start)
G.f.: Sum_{n>=1} n*a(n)*x^n/(n! + a(n)*x^n) = Sum_{n>=1} n^n*x^n/n!.
G.f.: Sum_{n>=1} n*a(n)*x^n/(n!*exp(nx) + a(n)*x^n) = x/(1-x).
Recurrence:
a(n) = n^(n-1) + (n-1)!*((-1)^n + Sum_{d|n, 1 1 with a(1)=1.
(End)

A354277 Product_{n>=1} 1 / (1 - x^n/n!)^a(n) = exp(-x) / (1 - x).

Original entry on oeis.org

0, 1, 2, 3, 24, 70, 720, 4305, 39200, 337176, 3628800, 38417610, 479001600, 6128488080, 87104969952, 1297383162075, 20922789888000, 354250929192160, 6402373705728000, 121407227453840328, 2432849766865689600, 51041047393559059200, 1124000727777607680000
Offset: 1

Views

Author

Ilya Gutkovskiy, May 22 2022

Keywords

Crossrefs

Cf. A000166, A006973, A137852, A353822, A354278.

Programs

  • Mathematica
    a[1] = 0; a[n_] := a[n] = (n - 1)! (1 - Sum[d d!^(-n/d) a[d], {d, Divisors[n]~Complement~{1, n}}]); Table[a[n], {n, 1, 23}]

Formula

a(1) = 0; a(n) = (n-1)! * (1 - Sum_{d|n, 1 < d < n} d * d!^(-n/d) * a(d)).
Previous Showing 11-20 of 23 results. Next

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