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 1-10 of 25 results. Next

A216688 Expansion of e.g.f. exp( x * exp(x^2) ).

Original entry on oeis.org

1, 1, 1, 7, 25, 121, 841, 4831, 40657, 325585, 2913841, 29910871, 301088041, 3532945417, 41595396025, 531109561711, 7197739614241, 100211165640481, 1507837436365537, 23123578483200295, 376697477235716281, 6348741961892933401, 111057167658053740201, 2032230051717594032767
Offset: 0

Views

Author

Joerg Arndt, Sep 14 2012

Keywords

Links

Crossrefs

Cf. A216507 (e.g.f. exp(x^2*exp(x))), A216689 (e.g.f. exp(x*exp(x)^2)).
Cf. A000248 (e.g.f. exp(x*exp(x))), A003725 (e.g.f. exp(x*exp(-x))).

Programs

  • Mathematica
    With[{nn = 25}, CoefficientList[Series[Exp[x Exp[x^2]], {x, 0, nn}], x] Range[0, nn]!] (* Bruno Berselli, Sep 14 2012 *)
  • PARI
    x='x+O('x^66);
Vec(serlaplace(exp( x * exp(x^2) )))
/* Joerg Arndt, Sep 14 2012 */
  • PARI
    a(n) = n!*sum(k=0, n\2, (n-2*k)^k/(k!*(n-2*k)!)); \\ Seiichi Manyama, Aug 18 2022

Formula

a(n) = n!*Sum_{m=floor((n+1)/2)..n} (2*m-n)^(n-m)/((2*m-n)!*(n-m)!). - Vladimir Kruchinin, Mar 09 2013
From Vaclav Kotesovec, Aug 06 2014: (Start)
a(n) ~ n^n / (r^n * exp((2*r^2*n)/(1+2*r^2)) * sqrt(3+2*r^2 - 2/(1 + 2*r^2))), where r is the root of the equation r*exp(r^2)*(1+2*r^2) = n.
(a(n)/n!)^(1/n) ~ exp(1/(3*LambertW(2^(1/3)*n^(2/3)/3))) * sqrt(2/(3*LambertW(2^(1/3)*n^(2/3)/3))).
(End)

A216507 E.g.f. exp( x^2 * exp(x) ).

Original entry on oeis.org

1, 0, 2, 6, 24, 140, 870, 5922, 45416, 381096, 3442410, 33382910, 345803172, 3801763836, 44156760830, 539962736250, 6929042527920, 93032248209872, 1303556965679826, 19018807375195638, 288341417011487420, 4534168069704168420, 73829219253218066022, 1242905562198878544626
Offset: 0

Views

Author

Joerg Arndt, Sep 14 2012

Keywords

Links

Crossrefs

Column k=2 of A292978.
Cf. A216688 (e.g.f. exp(x*exp(x^2))), A216689 (e.g.f. exp(x*exp(x)^2)).
Cf. A000248 (e.g.f. exp(x*exp(x))), A003725 (e.g.f. exp(x*exp(-x))).

Programs

  • Mathematica
    With[{nn = 25}, CoefficientList[Series[Exp[x^2 Exp[x]], {x, 0, nn}],
   x] Range[0, nn]!] (* Bruno Berselli, Sep 14 2012 *)
  • PARI
    x='x+O('x^66);
Vec(serlaplace(exp( x^2 * exp(x) )))
/* Joerg Arndt, Sep 14 2012 */

Formula

From Vaclav Kotesovec, Aug 06 2014: (Start)
a(n) ~ n^n / (exp(n*(1+r)/(2+r)) * r^n * sqrt((1+r)*(4+r)/(2+r))), where r is the root of the equation r^2*(2+r)*exp(r) = n.
(a(n)/n!)^(1/n) ~ exp(1/(3*LambertW(n^(1/3)/3))) / (3*LambertW(n^(1/3)/3)).
(End)
a(n) = Sum_{k = 0..n/2} C(n,2*k) * ((2*k)!/k!) * k^(n-2*k). - David Einstein, Oct 30 2016

A216689 Expansion of e.g.f. exp( x * exp(x)^2 ).

Original entry on oeis.org

1, 1, 5, 25, 153, 1121, 9373, 87417, 898033, 10052353, 121492341, 1573957529, 21729801481, 318121178337, 4917743697805, 79981695655801, 1364227940101857, 24335561350365953, 452874096174214117, 8772713803852981785, 176541611843378273401, 3684142819311127955041, 79596388271096140589949
Offset: 0

Views

Author

Joerg Arndt, Sep 14 2012

Keywords

Links

Crossrefs

Cf. A216507 (e.g.f. exp(x^2*exp(x))), A216688 (e.g.f. exp(x*exp(x^2))).
Cf. A000248 (e.g.f. exp(x*exp(x))), A003725 (e.g.f. exp(x*exp(-x))).
Cf. A240165 (e.g.f. exp(x*(1+exp(x)^2))).

Programs

  • Mathematica
    With[{nn = 25}, CoefficientList[Series[Exp[x Exp[x]^2], {x, 0, nn}], x] Range[0, nn]!] (* Bruno Berselli, Sep 14 2012 *)
  • PARI
    x='x+O('x^66);
Vec(serlaplace(exp( x * exp(x)^2 )))
/* Joerg Arndt, Sep 14 2012 */
  • PARI
    /* From o.g.f.: */
{a(n)=local(A=1);A=sum(k=0, n, x^k/(1 - 2*k*x +x*O(x^n))^(k+1));polcoeff(A, n)}
for(n=0,25,print1(a(n),", ")) /* Paul D. Hanna, Aug 02 2014 */
  • PARI
    /* From binomial sum: */
{a(n)=sum(k=0,n, binomial(n,k)*(2*k)^(n-k))}
for(n=0,30,print1(a(n),", ")) /* Paul D. Hanna, Aug 02 2014 */

Formula

O.g.f.: Sum_{n>=0} x^n / (1 - 2*n*x)^(n+1). - Paul D. Hanna, Aug 02 2014
a(n) = Sum_{k=0..n} binomial(n,k) * (2*k)^(n-k) for n>=0. - Paul D. Hanna, Aug 02 2014
From Vaclav Kotesovec, Aug 06 2014: (Start)
a(n) ~ n^n / (exp(2*n*r/(1+2*r)) * r^n * sqrt((1+6*r+4*r^2)/(1+2*r))), where r is the root of the equation r*(1+2*r)*exp(2*r) = n.
(a(n)/n!)^(1/n) ~ exp(1/(2*LambertW(sqrt(n/2)))) / LambertW(sqrt(n/2)).
(End)

A292952 E.g.f.: exp(-x * exp(x)).

Original entry on oeis.org

1, -1, -1, 2, 9, 4, -95, -414, 49, 10088, 55521, 13870, -2024759, -15787188, -28612415, 616876274, 7476967905, 32522642896, -209513308607, -4924388011050, -38993940088199, -11731860520780, 3807154270837281, 52018152493218010, 278413297030360273
Offset: 0

Views

Author

Seiichi Manyama, Sep 27 2017

Keywords

Links

Crossrefs

Column k=1 of A293015.
Cf. this sequence (k=1), A292953 (k=2), A292954 (k=3), A292955 (k=4).
Cf. A003725.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[Exp[-x Exp[x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 15 2023 *)
  • PARI
    x='x+O('x^66); Vec(serlaplace(exp(-x*exp(x))))

Formula

a(n) = (-1)^n * A003725(n).

A292948 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where A(0,k) = 1 and A(n,k) = (-1)^(k+1) * Sum_{i=0..n-1} (-1)^i * binomial(n-1,i) * binomial(i+1,k) * A(n-1-i,k) for n > 0.

Original entry on oeis.org

1, 1, -1, 1, 1, 2, 1, 0, -1, -5, 1, 0, 1, -2, 15, 1, 0, 0, -3, 9, -52, 1, 0, 0, 1, 9, -4, 203, 1, 0, 0, 0, -4, -40, -95, -877, 1, 0, 0, 0, 1, 10, 210, 414, 4140, 1, 0, 0, 0, 0, -5, -10, -1176, 49, -21147, 1, 0, 0, 0, 0, 1, 15, -105, 7273, -10088, 115975, 1, 0, 0
Offset: 0

Views

Author

Seiichi Manyama, Sep 27 2017

Keywords

Examples

			Square array begins:
    1,  1,  1,  1, 1, ...
   -1,  1,  0,  0, 0, ...
    2, -1,  1,  0, 0, ...
   -5, -2, -3,  1, 0, ...
   15,  9,  9, -4, 1, ...

Links

Crossrefs

Columns k=0-5 give: A292935, A003725, A292909, A292910, A292912, A292950.
Rows n=0 gives A000012.
Main diagonal gives A000012.
Cf. A145460.

Programs

  • Ruby
    def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(k, n)
  ary = [1]
  (1..n).each{|i| ary << (-1) ** (k % 2 + 1) * (0..i - 1).inject(0){|s, j| s + (-1) ** (j % 2) * ncr(i - 1, j) * ncr(j + 1, k) * ary[i - 1 - j]}}
  ary
end
def A292948(n)
  a = []
  (0..n).each{|i| a << A(i, n - i)}
  ary = []
  (0..n).each{|i|
    (0..i).each{|j|
      ary << a[i - j][j]
    }
  }
  ary
end
p A292948(20)

A383991 Series expansion of the exponential generating function exp(-tridend(-x)) - 1 where tridend(x) = (1 - 3*x - sqrt(1-6*x+x^2)) / (4*x) (A001003).

Original entry on oeis.org

0, 1, -5, 49, -743, 15421, -407909, 13135165, -498874991, 21838772377, -1082819193029, 59983280191561, -3671752681190615, 246130081055714389, -17932045676505509093, 1410893903131294766101, -119227840965746009631839, 10769985399394862863318705
Offset: 0

Views

Author

Michael De Vlieger, May 16 2025

Keywords

Comments

The series -tridend(-x) is the inverse for the substitution of the series trias(x), given by the suspension of the Koszul dual of trias. - Bérénice Delcroix-Oger, May 28 2025

Links

Crossrefs

Cf. A003725, A097388, A111884, A112242, A177885, A318215, A383987, A383990, A383992, A383993, A383994, A383995.

Programs

  • Mathematica
    nn = 19; f[x_] := Exp[x] - 1;
Range[0, nn]! * CoefficientList[Series[f[(1 + 3*x - Sqrt[1 + 6*x + x^2])/(4*x)], {x, 0, nn}], x]

A383995 Series expansion of the exponential generating function exp(ff6^!(x)) - 1 where ff6^!(x) = x * (1-3*x-x^2+x^3) / (1+3*x+x^2-x^3).

Original entry on oeis.org

0, 1, -11, 61, -215, -1559, 62941, -1371131, 26310481, -474554735, 7824076741, -98881279859, -176260664711, 87457412423161, -5077434546358355, 234510433823788501, -10016559114085864799, 413333665704129673249, -16704968283664639137899, 660340818239784197391325
Offset: 0

Views

Author

Michael De Vlieger, May 16 2025

Keywords

Comments

The series ff6^!(x) is the inverse for the substitution of the series ff6(x) (given by A231690), given by the suspension of the Koszul dual of FF6. - Bérénice Delcroix-Oger, May 28 2025

Links

Crossrefs

Cf. A003725, A097388, A111884, A112242, A177885, A318215, A383989, A383990, A383991, A383992, A383993, A383994.

Programs

  • Mathematica
    nn = 19; f[x_] := Exp[x] - 1;
Range[0, nn]! * CoefficientList[Series[f[x*(1 - 3*x - x^2 + x^3)/(1 + 3*x + x^2 - x^3)], {x, 0, nn}], x]

A008405 n-th derivative of x^(1/x) at x=1.

Original entry on oeis.org

1, 1, -2, 3, 4, -90, 786, -5670, 34784, -136584, -824760, 33137280, -633666648, 10089623544, -145675230960, 1910939579640, -21215723677440, 136130901474240, 2280768466608576, -135531682778927808, 4380044490023909760, -119144344839822570240
Offset: 0

Views

Author

Olivier Gérard

Keywords

Links

Programs

  • Mathematica
    Function[ n, Series[ (1+x)^(1/(1+x)), {x, 0, n} ]//(Table[ SeriesCoefficient[ #, i ]*i!, {i, 0, n} ])& ][ 20 ]
a[n_] := Sum[ StirlingS1[n, k]*Sum[ (-j)^(k-j)*Binomial[k, j], {j, 0, k}], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Nov 28 2012, after Vladeta Jovovic *)
NestList[Factor[D[#1, x]] &, x^(1/x), 22] /. (x -> 1) (* Robert G. Wilson v, Feb 03 2013 *)

Formula

a(n) = Sum_{k=0..n} Stirling1(n, k)*A003725(k). - Vladeta Jovovic, Oct 02 2003

A292973 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) = 1 and T(n,k) = (-1)^(k+1) * k! * Sum_{i=0..n-1} (-1)^i * binomial(n-1,i) * binomial(i+1,k) * T(n-1-i,k) for n > 0.

Original entry on oeis.org

1, 1, -1, 1, 1, 2, 1, 0, -1, -5, 1, 0, 2, -2, 15, 1, 0, 0, -6, 9, -52, 1, 0, 0, 6, 24, -4, 203, 1, 0, 0, 0, -24, -140, -95, -877, 1, 0, 0, 0, 24, 60, 870, 414, 4140, 1, 0, 0, 0, 0, -120, 240, -5922, 49, -21147, 1, 0, 0, 0, 0, 120, 360, -4830, 45416, -10088, 115975
Offset: 0

Views

Author

Seiichi Manyama, Sep 27 2017

Keywords

Examples

			Square array begins:
   1,  1,  1,   1,  1, ...
  -1,  1,  0,   0,  0, ...
   2, -1,  2,   0,  0, ...
  -5, -2, -6,   6,  0, ...
  15,  9, 24, -24, 24, ...

Links

Crossrefs

Columns k=0-5 give: A292935, A003725, A292907, A292908, A292969, A292970.
Rows n=0 gives A000012.
Main diagonal gives A000142.
Cf. A292948, A292978.

Programs

  • Ruby
    def f(n)
  return 1 if n < 2
  (1..n).inject(:*)
end
def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(k, n)
  ary = [1]
  (1..n).each{|i| ary << (-1) ** (k % 2 + 1) * f(k) * (0..i - 1).inject(0){|s, j| s + (-1) ** (j % 2) * ncr(i - 1, j) * ncr(j + 1, k) * ary[i - 1 - j]}}
  ary
end
def A292973(n)
  a = []
  (0..n).each{|i| a << A(i, n - i)}
  ary = []
  (0..n).each{|i|
    (0..i).each{|j|
      ary << a[i - j][j]
    }
  }
  ary
end
p A292973(20)

Formula

T(n,k) = n! * Sum_{j=0..floor(n/k)} (-j)^(n-k*j)/(j! * (n-k*j)!) for k > 0. - Seiichi Manyama, Jul 10 2022

A383992 Series expansion of the exponential generating function exp(arbustive(x)) - 1 where arbustive(x) = (log(1+x) - x^2) / (1+x).

Original entry on oeis.org

0, 1, -4, 3, 40, -330, 1626, -3150, -54592, 1060920, -13022280, 127171440, -889086648, -283184616, 179750627616, -4895777544840, 99124001788800, -1721513264431680, 25736021675994816, -292896125040673728, 639149345262276480, 106178474282318726400
Offset: 0

Views

Author

Michael De Vlieger, May 16 2025

Keywords

Links

Crossrefs

Cf. A003725, A097388, A111884, A112242, A114285, A177885, A318215, A383990, A383991, A383993, A383994, A383995.

Programs

  • Mathematica
    nn = 21; f[x_] := Exp[x] - 1;
Range[0, nn]! * CoefficientList[Series[f[(Log[1 + x] - x^2)/(1 + x)], {x, 0, nn}], x]
Showing 1-10 of 25 results. Next

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