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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A336950 E.g.f.: 1 / (1 - x * exp(2*x)).

Original entry on oeis.org

1, 1, 6, 42, 392, 4600, 64752, 1063216, 19952256, 421227648, 9880951040, 254960721664, 7176891675648, 218857588139008, 7187394935347200, 252897556424140800, 9491754142468702208, 378509920569294684160, 15982018774576565649408, 712306819507400060502016
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 08 2020

Keywords

Links

Crossrefs

Column k=2 of A351790.
Cf. A001787, A006153, A122704, A216794, A235328, A336947, A336951, A336952.

Programs

  • Mathematica
    nmax = 19; CoefficientList[Series[1/(1 - x Exp[2 x]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[n! Sum[(2 (n - k))^k/k!, {k, 0, n}], {n, 1, 19}]]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] k 2^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]
  • PARI
    seq(n)={ Vec(serlaplace(1 / (1 - x*exp(2*x + O(x^n))))) } \\ Andrew Howroyd, Aug 08 2020

Formula

a(n) = n! * Sum_{k=0..n} (2 * (n-k))^k / k!.
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * k * 2^(k-1) * a(n-k).
a(n) ~ n! * (2/LambertW(2))^n / (1 + LambertW(2)). - Vaclav Kotesovec, Aug 09 2021

A364987 E.g.f. satisfies A(x) = 1 + x*exp(x)*A(x)^4.

Original entry on oeis.org

1, 1, 10, 183, 5140, 196005, 9468486, 554425963, 38171336680, 3022130473065, 270537702834250, 27021535857472431, 2979254055371578524, 359411244032212931533, 47093111659782104431438, 6660135357832421444841555, 1011181346455643980818939856
Offset: 0

Views

Author

Seiichi Manyama, Aug 15 2023

Keywords

Crossrefs

Cf. A006153, A295238, A364983.
Cf. A002293, A349331.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*binomial(4*k, k)/((3*k+1)*(n-k)!));

Formula

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(4*k+1,k)/( (4*k+1)*(n-k)! ) = n! * Sum_{k=0..n} k^(n-k) * A002293(k)/(n-k)!.
a(n) ~ 2*sqrt(1 + LambertW(27/256)) * n^(n-1) / (3^(3/2) * exp(n) * LambertW(27/256)^n). - Vaclav Kotesovec, Nov 11 2024

A377530 Expansion of e.g.f. 1/(1 - x * exp(x))^3.

Original entry on oeis.org

1, 3, 18, 141, 1380, 16095, 217458, 3335745, 57225528, 1085066523, 22526087070, 508042140573, 12367076890644, 323130848000727, 9018976230237834, 267789942962863065, 8427492557547704688, 280194087519310655667, 9813332205452943323190, 361109786425470021564021
Offset: 0

Views

Author

Seiichi Manyama, Oct 30 2024

Keywords

Crossrefs

Cf. A006153, A377529.
Cf. A377532, A377534.
Cf. A377504.

Programs

  • Mathematica
    nmax=19; CoefficientList[Series[1/(1 - x * Exp[x])^3,{x,0,nmax}],x]Range[0,nmax]! (* Stefano Spezia, Feb 05 2025 *)
  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*binomial(k+2, 2)/(n-k)!);

Formula

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(k+2,2)/(n-k)!.
a(n) ~ n! * n^2 / (2 * (1+LambertW(1))^3 * LambertW(1)^n). - Vaclav Kotesovec, Oct 31 2024

A380015 Expansion of e.g.f. 1/sqrt(1 - 2*x*exp(x)).

Original entry on oeis.org

1, 1, 5, 36, 361, 4640, 72771, 1347598, 28778849, 696288888, 18823644595, 562350743306, 18397666000209, 654164843763340, 25118967828553067, 1035914449832324070, 45665488606439586241, 2142825945301659242576, 106641225471890568771747, 5610282675990428302440130
Offset: 0

Views

Author

Seiichi Manyama, Jan 09 2025

Keywords

Crossrefs

Cf. A006153, A380017, A380019.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (-2)^k*k^(n-k)*binomial(-1/2, k)/(n-k)!);

Formula

a(n) = n! * Sum_{k=0..n} (-2)^k * k^(n-k) * binomial(-1/2,k)/(n-k)!.
a(n) ~ sqrt(2) * n^n / (sqrt(1 + LambertW(1/2)) * exp(n) * LambertW(1/2)^n). - Vaclav Kotesovec, Jan 23 2025

A380017 Expansion of e.g.f. 1/(1 - 3*x*exp(x))^(1/3).

Original entry on oeis.org

1, 1, 6, 55, 716, 12085, 250726, 6172915, 175903400, 5694587209, 206438732810, 8284550317351, 364605758728828, 17461047965591581, 903964982917764782, 50306323769422679995, 2994799872257498255696, 189906103853462927405329, 12779300537432602189228306
Offset: 0

Views

Author

Seiichi Manyama, Jan 09 2025

Keywords

Crossrefs

Cf. A006153, A380015, A380019.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (-3)^k*k^(n-k)*binomial(-1/3, k)/(n-k)!);

Formula

a(n) = n! * Sum_{k=0..n} (-3)^k * k^(n-k) * binomial(-1/3,k)/(n-k)!.
a(n) ~ sqrt(2*Pi) * n^(n - 1/6) / (Gamma(1/3) * (1 + LambertW(1/3))^(1/3) * exp(n) * LambertW(1/3)^n). - Vaclav Kotesovec, Jan 23 2025

A364983 E.g.f. satisfies A(x) = 1 + x*exp(x)*A(x)^3.

Original entry on oeis.org

1, 1, 8, 111, 2332, 66125, 2368086, 102616759, 5222638856, 305436798009, 20186656927210, 1488021110087171, 121044207712073196, 10771321471267219525, 1040877104088653696606, 108549742436141933697135, 12151467262433697322437136, 1453367472748861203540942065
Offset: 0

Views

Author

Seiichi Manyama, Aug 15 2023

Keywords

Crossrefs

Cf. A364984, A364985, A364986.
Cf. A006153, A295238, A364987.
Cf. A001764, A307678.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*binomial(3*k, k)/((2*k+1)*(n-k)!));

Formula

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(3*k+1,k)/( (3*k+1)*(n-k)! ) = n! * Sum_{k=0..n} k^(n-k) * A001764(k)/(n-k)!.
a(n) ~ sqrt(3) * sqrt(1 + LambertW(4/27)) * n^(n-1) / (2^(3/2) * exp(n) * LambertW(4/27)^n). - Vaclav Kotesovec, Nov 11 2024

A364980 E.g.f. satisfies A(x) = 1 + x*A(x)*exp(x*A(x)^2).

Original entry on oeis.org

1, 1, 4, 33, 412, 6945, 147846, 3807601, 115151464, 4001162913, 157096369450, 6878742553881, 332361857826780, 17566215943990753, 1008161606338206334, 62440146891413434305, 4151012174991960338896, 294834882756167048975553
Offset: 0

Views

Author

Seiichi Manyama, Aug 15 2023

Keywords

Crossrefs

Cf. A006153, A161633, A161635, A364981.

Programs

  • Mathematica
    Join[{1}, Table[n! * Sum[k^(n-k) * Binomial[2*n-k+1,k] / ((2*n-k+1)*(n-k)!), {k,0,n}], {n,1,20}]] (* Vaclav Kotesovec, Nov 18 2023 *)
  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*binomial(2*n-k+1, k)/((2*n-k+1)*(n-k)!));

Formula

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(2*n-k+1,k)/( (2*n-k+1)*(n-k)! ).
a(n) ~ sqrt((1 + r*s^2)/(6 + 4*r*s^2)) * n^(n-1) / (exp(n) * r^(n + 1/2)), where r = 0.2190923703746024362724546703711998154573791458000... and s = 1.747404632046819382844696016554403302840973484745... are real roots of the system of equations 1 + exp(r*s^2)*r*s = s, 2*r*s^2*(s-1) = 1. - Vaclav Kotesovec, Nov 18 2023

A305306 Expansion of e.g.f. 1/(1 + log(1 - x)/(1 - x)).

Original entry on oeis.org

1, 1, 5, 35, 324, 3744, 51902, 839362, 15513096, 322550616, 7451677632, 189366303840, 5249764639248, 157666361452560, 5099445234111888, 176713626295062384, 6531995374500741888, 256537368987293878272, 10667901271715707803264, 468261481657502075856768, 21635865693957558515860224
Offset: 0

Views

Author

Ilya Gutkovskiy, May 29 2018

Keywords

Comments

a(n)/n! is the invert transform of [1, 1 + 1/2, 1 + 1/2 + 1/3, 1 + 1/2 + 1/3 + 1/4, ...].

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 35*x^3/3! + 324*x^4/4! + 3744*x^5/5! + 51902*x^6/6! + ...

Links

Crossrefs

Cf. A000254, A001008, A002805, A007840, A128044, A128045, A305307.
Cf. A006153, A087761.

Programs

  • Maple
    H:= proc(n) H(n):= 1/n +`if`(n=1, 0, H(n-1)) end:
b:= proc(n) option remember; `if`(n=0, 1,
      add(H(j)*b(n-j), j=1..n))
    end:
a:= n-> b(n)*n!:
seq(a(n), n=0..20);  # Alois P. Heinz, May 29 2018
  • Mathematica
    nmax = 20; CoefficientList[Series[1/(1 + Log[1 - x]/(1 - x)), {x, 0, nmax}], x] Range[0, nmax]!
nmax = 20; CoefficientList[Series[1/(1 - Sum[HarmonicNumber[k] x^k , {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[HarmonicNumber[k] a[n - k], {k, 1, n}]; Table[n! a[n], {n, 0, 20}]
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+log(1-x)/(1-x)))) \\ Seiichi Manyama, May 10 2023

Formula

E.g.f.: 1/(1 - Sum_{k>=1} (A001008(k)/A002805(k))*x^k).
a(n) ~ n! / ((1/LambertW(1)^2 - 1) * (1 - LambertW(1))^n). - Vaclav Kotesovec, Aug 08 2021
a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A006153(k). - Seiichi Manyama, May 10 2023

A305990 Expansion of e.g.f.: (1+x) / (exp(-x) - x).

Original entry on oeis.org

1, 3, 11, 58, 409, 3606, 38149, 470856, 6641793, 105398650, 1858413061, 36044759796, 762659322385, 17481598316742, 431535346662645, 11413394655983536, 321989729198400385, 9651573930139850610, 306321759739045148293, 10262156907184058219340
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 16 2018

Keywords

Crossrefs

Cf. A305133, A006153, A009306, A009444, A072597, A089148, A302397.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1+x)/(E^(-x)-x), {x, 0, nmax}], x] * Range[0, nmax]!
a={1};For[n=1,n<20,n++,AppendTo[a,Sum[(n!)*((n-k+1)^(k-1))*(n+1)/(k!),{k,0,n+1}]]]; a (* Detlef Meya, Sep 05 2023 *)

Formula

a(n) ~ n! / LambertW(1)^(n+1).
a(n) = (-1)^n * A009444(n+1).
a(n) = Sum_{k=0..n+1} (n+1)!*(n-k+1)^(k-1)/k! for n > 0. - Detlef Meya, Sep 05 2023

A336951 E.g.f.: 1 / (1 - x * exp(3*x)).

Original entry on oeis.org

1, 1, 8, 69, 780, 11145, 191178, 3823785, 87406056, 2247785073, 64228084110, 2018771719569, 69221032558956, 2571290056399545, 102860527370221026, 4408690840306136505, 201557641172689004112, 9790792086366911655009, 503570143277542340304534
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 08 2020

Keywords

Links

Crossrefs

Column k=3 of A351790.
Cf. A006153, A027471, A235328, A255927, A328182, A336948, A336950, A336952.

Programs

  • Mathematica
    nmax = 18; CoefficientList[Series[1/(1 - x Exp[3 x]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[n! Sum[(3 (n - k))^k/k!, {k, 0, n}], {n, 1, 18}]]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] k 3^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
  • PARI
    seq(n)={ Vec(serlaplace(1 / (1 - x*exp(3*x + O(x^n))))) } \\ Andrew Howroyd, Aug 08 2020

Formula

a(n) = n! * Sum_{k=0..n} (3 * (n-k))^k / k!.
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * k * 3^(k-1) * a(n-k).
a(n) ~ n! * (3/LambertW(3))^n / (1 + LambertW(3)). - Vaclav Kotesovec, Aug 09 2021
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