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.

Showing 1-9 of 9 results.

A298374 Expansion of e.g.f. 1/(1 - x)^exp(-x).

Original entry on oeis.org

1, 1, 0, 0, 6, 15, 65, 595, 4004, 32865, 322307, 3316511, 37845214, 471644173, 6319617369, 91114344217, 1404670896264, 23050054222177, 401305630237239, 7387282161642715, 143360257370842146, 2925289119525173741, 62612350725688075941, 1402681525332544374325
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 18 2018

Keywords

Comments

Exponential transform of A002741.

Examples

			1/(1 - x)^exp(-x) = 1 + x/1! + 6*x^4/4! + 15*x^5/5! + 65*x^6/6! + 595*x^7/7! + ...

Links

Crossrefs

Cf. A002741, A007116, A009198, A191365.

Programs

  • Maple
    a:=series(1/(1-x)^exp(-x),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 26 2019
  • Mathematica
    nmax = 23; CoefficientList[Series[1/(1 - x)^Exp[-x], {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x)^exp(-x))) \\ Seiichi Manyama, May 03 2022

Formula

a(n) ~ n! * n^(exp(-1)-1) / Gamma(exp(-1)). - Vaclav Kotesovec, May 04 2018

A007116 Expansion of e.g.f. (1+x)^(exp(x)).

Original entry on oeis.org

1, 1, 2, 6, 18, 75, 295, 1575, 7196, 48993, 230413, 2164767, 8055938, 139431149, 70125991, 14201296057, -77573062280, 2389977322593, -28817693086263, 615493949444827, -10403976760589602, 215611836994976237
Offset: 0

Views

Author

Simon Plouffe

Keywords

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002741, A009198, A191365.

Programs

  • Mathematica
    With[{nn=25},CoefficientList[Series[(1+x)^Exp[x],{x,0,nn}], x] Range[ 0,nn]!] (* Harvey P. Dale, Sep 21 2011 *)
  • Maxima
    a(n):=sum(sum(binomial(n,i)*k^i*stirling1(n-i,k),i,0,n-k),k,1,n); /* Vladimir Kruchinin, Jun 01 2011 */

Formula

a(n) = sum(k=1..n, sum(i=0..n-k, binomial(n,i)*k^i*Stirling1(n-i,k))), n>0, a(0)=1. - Vladimir Kruchinin, Jun 01 2011
|a(n)| ~ n!/(Gamma(-exp(-1))*n^(1+exp(-1))). - Vaclav Kotesovec, Jun 27 2013

Extensions

Definition and terms corrected, and more terms added by Joerg Arndt, Jun 01 2011

A305407 Expansion of e.g.f. 1/(1 + log(1 - x)*exp(x)).

Original entry on oeis.org

1, 1, 5, 32, 274, 2939, 37833, 568210, 9753280, 188342949, 4041170695, 95380234366, 2455830637412, 68501591450447, 2057726452045145, 66227424015265178, 2273614433910697920, 82932491842062712873, 3202994529476330549163, 130577628147690206429038, 5603479009890212632226756
Offset: 0

Views

Author

Ilya Gutkovskiy, May 31 2018

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 32*x^3/3! + 274*x^4/4! + 2939*x^5/5! + 37833*x^6/6! + ...

Links

Crossrefs

Cf. A002104, A006153, A007840, A009324, A191365.

Programs

  • Maple
    a:=series(1/(1+log(1-x)*exp(x)),x=0,21): seq(n!*coeff(a,x,n),n=0..20); # Paolo P. Lava, Mar 26 2019
  • Mathematica
    nmax = 20; CoefficientList[Series[1/(1 + Log[1 - x] Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[HypergeometricPFQ[{1, 1, 1 - k}, {2}, -1] a[n - k]/(k - 1)!, {k, 1, n}]; Table[n! a[n], {n, 0, 20}]

Formula

a(n) ~ n! / ((1 + exp(r)/r) * (1 - exp(-r))^(n+1)), where r = 0.62747017959751658496114808922921433658821962606026068561095... is the root of the equation r*exp(1 - exp(-r)) = 1. - Vaclav Kotesovec, Mar 26 2019
a(0) = 1; a(n) = Sum_{k=1..n} A002104(k) * binomial(n,k) * a(n-k). - Seiichi Manyama, May 04 2022

A353995 Expansion of e.g.f. 1/(1 - x)^(exp(x) - 1).

Original entry on oeis.org

1, 0, 2, 6, 30, 185, 1315, 10682, 97692, 991797, 11060413, 134368344, 1766007122, 24963786003, 377633418279, 6086719267852, 104134471945368, 1884698592318537, 35976835400864745, 722386383476096128, 15220456179011671358, 335769403850849998439
Offset: 0

Views

Author

Seiichi Manyama, May 13 2022

Keywords

Links

Crossrefs

Cf. A052863, A191365, A353994.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[1/(1-x)^(Exp[x]-1),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 03 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x)^(exp(x)-1)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-log(1-x)*(exp(x)-1))))

Formula

E.g.f.: exp( -log(1 - x) * (exp(x) - 1) ).
a(0) = 1; a(n) = Sum_{k=1..n} A052863(k) * binomial(n-1,k-1) * a(n-k).
a(n) ~ n! * n^(exp(1)-2) / Gamma(exp(1)-1) * (1 - (exp(1)-2)*exp(1)*log(n)/n). - Vaclav Kotesovec, May 13 2022

A351881 Expansion of e.g.f. 1 / (1 - x)^cosh(x).

Original entry on oeis.org

1, 1, 2, 9, 42, 235, 1605, 12446, 108836, 1061565, 11402565, 133806134, 1703059974, 23366177055, 343788954691, 5399655967642, 90173526680152, 1595513146457993, 29817130502252169, 586883850601630054, 12135450890055396810, 263012688923611212107, 5962155058714267567319
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 23 2022

Keywords

Crossrefs

Cf. A009193, A009231, A009416, A191365, A351880.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[1/(1 - x)^Cosh[x], {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(1/(1-x)^cosh(x))) \\ Michel Marcus, Feb 23 2022

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * |A009416(k)| * a(n-k).
a(n) ~ n! * n^(cosh(1)-1) / Gamma(cosh(1)). - Vaclav Kotesovec, Feb 23 2022

A356925 E.g.f. satisfies A(x) = 1/(1 - x)^(exp(x) * A(x)).

Original entry on oeis.org

1, 1, 6, 51, 614, 9655, 188209, 4389532, 119363488, 3711190881, 129932611723, 5060364817200, 217054300138136, 10168837756846145, 516709033266165479, 28306732060349788908, 1663231006737554997168, 104344911495734048046929
Offset: 0

Views

Author

Seiichi Manyama, Sep 04 2022

Keywords

Links

Crossrefs

Cf. A191365, A356926.
Cf. A356752, A356753.
Cf. A000272, A002104, A356927.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[LambertW[E^x * Log[1-x]]/(E^x * Log[1-x]), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 14 2022 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(-exp(x)*log(1-x))^k/k!)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(exp(x)*log(1-x)))))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(lambertw(exp(x)*log(1-x))/(exp(x)*log(1-x))))

Formula

E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (-exp(x) * log(1-x))^k / k!.
E.g.f.: A(x) = exp( -LambertW(exp(x) * log(1-x)) ).
E.g.f.: A(x) = LambertW(exp(x) * log(1-x))/(exp(x) * log(1-x)).
a(n) ~ sqrt(1 + exp(1+r)/(1-r)) * n^(n-1) / (r^(n - 1/2) * exp(n-1)), where r = 0.249272970940807862774650581662931601739615720771408527... is the root of the equation exp(1+r) * log(1-r) = -1. - Vaclav Kotesovec, Nov 14 2022

A308332 a(n) = n! * [x^n] 1/(1 - x)^exp(n*x).

Original entry on oeis.org

1, 1, 6, 60, 936, 21495, 681480, 28157451, 1455590528, 91689831225, 6907344210400, 612700433073707, 63107430169208832, 7455570223877314721, 999839697339310324224, 150885818035154310155625, 25434297819615665229168640, 4758031551536565527014516561
Offset: 0

Views

Author

Ilya Gutkovskiy, May 20 2019

Keywords

Crossrefs

Cf. A190314, A191365, A308333.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[1/(1 - x)^Exp[n x], {x, 0, n}], {n, 0, 17}]

A345454 E.g.f.: log(1 - log(1 - x) * exp(x)).

Original entry on oeis.org

0, 1, 2, 1, -5, 3, 141, 348, -1938, 3013, 274327, 1583338, -4613476, 41135339, 3201505997, 33153080054, 49123558416, 2360520208825, 133442956587099, 2109709010976874, 14751973018988252, 338170133891984663, 15120630911878380457, 324654726628159335686
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 18 2021

Keywords

Links

Crossrefs

Cf. A002104, A009321, A009324, A191365, A346427.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[Log[1 - Log[1 - x] Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
A002104[n_] := A002104[n] = n! Sum[1/((n - k) k!), {k, 0, n - 1}]; a[0] = 0; a[n_] := a[n] = A002104[n] - (1/n) Sum[Binomial[n, k] A002104[n - k] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 23}]
  • PARI
    my(x='x+O('x^25)); concat(0, Vec(serlaplace(log(1 - log(1 - x) * exp(x))))) \\ Michel Marcus, Jul 19 2021

Formula

a(0) = 0; a(n) = A002104(n) - (1/n) * Sum_{k=1..n-1} binomial(n,k) * A002104(n-k) * k * a(k).

A356926 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^exp(x).

Original entry on oeis.org

1, 1, 2, 3, 10, 35, 121, 1092, 5216, 39321, 558643, 2433508, 48144944, 688652549, 2176310995, 145742587616, 1334993574032, 5551320939809, 799648465754835, 1049695714507276, 90069170433616208, 6281942689646504501, -53282051261767839293, 2356158301117802408472
Offset: 0

Views

Author

Seiichi Manyama, Sep 04 2022

Keywords

Links

Crossrefs

Cf. A191365, A356925.
Cf. A356912, A356913.
Cf. A002104, A177885.
Cf. A141209.

Programs

  • Mathematica
    nmax = 23; A[_] = 1;
Do[A[x_] = ((1 - x)^(-Exp[x]))^(1/A[x]) + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(-exp(x)*log(1-x))^k/k!)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-exp(x)*log(1-x)))))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(-exp(x)*log(1-x)/lambertw(-exp(x)*log(1-x))))

Formula

E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (-exp(x) * log(1-x))^k / k!.
E.g.f.: A(x) = exp( LambertW(-exp(x) * log(1-x)) ).
E.g.f.: A(x) = -exp(x) * log(1-x)/LambertW(-exp(x) * log(1-x)).
Showing 1-9 of 9 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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