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 1-5 of 5 results.

A355378 Expansion of e.g.f. exp(exp(3*x) - exp(x)).

Original entry on oeis.org

1, 2, 12, 82, 688, 6754, 75096, 928386, 12591392, 185384130, 2938319144, 49799613538, 897495547184, 17118975292514, 344206910941624, 7270287035936706, 160826794265399360, 3716047107259486082, 89472755268582494792, 2240097688067896960674, 58207872357772581544272
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 30 2022

Keywords

Links

Crossrefs

Cf. A143405, A355291, A355379, A355381.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Exp[Exp[3*x] - Exp[x]], {x, 0, nmax}], x] * Range[0, nmax]!
Table[Sum[Binomial[n,k] * 3^k * BellB[k] * BellB[n-k, -1], {k, 0, n}], {n, 0, 20}]
  • PARI
    my(x='x+O('x^25)); Vec(serlaplace(exp(exp(3*x) - exp(x)))) \\ Michel Marcus, Jun 30 2022

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * 3^k * Bell(k) * Bell(n-k, -1).
a(0) = 1; a(n) = Sum_{k=1..n} (3^k - 1) * binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Jun 30 2022
a(n) ~ exp(exp(3*z) - exp(z) - n) * (n/z)^(n + 1/2) / sqrt(3*(1 + 3*z)*exp(3*z) - (1 + z)*exp(z)), where z = LambertW(n)/3 - 1/(1 + 3/LambertW(n) - 9 * n^(2/3) * (1 + LambertW(n)) / LambertW(n)^(5/3)). - Vaclav Kotesovec, Jul 03 2022
a(n) ~ (3*n/LambertW(n))^n * exp(n/LambertW(n) - (n/LambertW(n))^(1/3) - n) / sqrt(1 + LambertW(n)). - Vaclav Kotesovec, Jul 10 2022

A355380 Expansion of e.g.f. exp(exp(3*x) + exp(2*x) - 2).

Original entry on oeis.org

1, 5, 38, 355, 3879, 48050, 661163, 9961745, 162598044, 2851150665, 53350521523, 1059447004560, 22224898346989, 490589320542305, 11356591577861398, 274886065370874775, 6939205217774546339, 182273695066097752170, 4971724931587003394863, 140559648864263508395965
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 30 2022

Keywords

Links

Crossrefs

Cf. A143405, A355291, A355381.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Exp[Exp[3*x] + Exp[2*x] - 2], {x, 0, nmax}], x] * Range[0, nmax]!
Table[Sum[Binomial[n,k] * 3^k * 2^(n-k) * BellB[k] * BellB[n-k], {k, 0, n}], {n, 0, 20}]
  • PARI
    my(x='x+O('x^25)); Vec(serlaplace(exp(exp(3*x) + exp(2*x) - 2))) \\ Michel Marcus, Jun 30 2022

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * 3^k * 2^(n-k) * Bell(k) * Bell(n-k).
a(0) = 1; a(n) = Sum_{k=1..n} (3^k + 2^k) * binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Jun 30 2022
a(n) ~ exp(exp(3*z) + exp(2*z) - 2 - n) * (n/z)^(n + 1/2) / sqrt(3*(1 + 3*z)*exp(3*z) + 2*(1 + 2*z)*exp(2*z)), where z = LambertW(n)/3 - 1/(2 + 3/LambertW(n) + 9 * n^(1/3) * (1 + LambertW(n)) / (2*LambertW(n)^(4/3))). - Vaclav Kotesovec, Jul 03 2022

A355409 Expansion of e.g.f. 1/(1 + exp(2*x) - exp(3*x)).

Original entry on oeis.org

1, 1, 7, 55, 571, 7471, 117307, 2148175, 44958571, 1058555791, 27693129307, 796934764495, 25018548004171, 850870651904911, 31163746960955707, 1222922731101304015, 51189052318085027371, 2276586205163067346831, 107204914362429152404507
Offset: 0

Views

Author

Seiichi Manyama, Jul 01 2022

Keywords

Comments

Conjecture: Let k be a positive integer. The sequence obtained by reducing a(n) modulo k is eventually periodic with period dividing phi(k) = A000010(k). For example, modulo 9 we obtain the sequence [1, 1, 7, 1, 4, 1, 1, 1, 7, 1, 4, 1, 1, 1, 7, 1, 4, 1, 1, ...] with an apparent period of 6 = phi(9) beginning at a(1). Cf. A354242. - Peter Bala, Apr 16 2024

Crossrefs

Cf. A371460 (binomial transform).
Cf. A000010, A354242, A355381, A355408, A370092.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+exp(2*x)-exp(3*x))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, (3^j-2^j)*binomial(i, j)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} (3^k - 2^k) * binomial(n,k) * a(n-k).
a(n) ~ n! / ((3 + r^2) * log(r)^(n+1)), where r = (1 + 2*cosh(log((29 + 3*sqrt(93))/2)/3))/3. - Vaclav Kotesovec, Jul 01 2022

A355398 Expansion of e.g.f. exp(exp(3*x)/3 - exp(2*x)/2 + 1/6).

Original entry on oeis.org

1, 0, 1, 5, 22, 115, 761, 5880, 49897, 460045, 4621366, 50385555, 590795217, 7389964400, 98105330961, 1377426850805, 20388005470582, 317112889169555, 5167636268318921, 88001180739368680, 1562559584723343417, 28871671817796197885, 554116841783123679446
Offset: 0

Views

Author

Seiichi Manyama, Jun 30 2022

Keywords

Crossrefs

Cf. A355381.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Exp[Exp[3*x]/3 - Exp[2*x]/2 + 1/6], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 30 2022 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(exp(3*x)/3-exp(2*x)/2+1/6)))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, (3^(j-1)-2^(j-1))*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} (3^(k-1) - 2^(k-1)) * binomial(n-1,k-1) * a(n-k).
a(n) ~ exp(exp(3*r)/3 - exp(2*r)/2 + 1/6 - n) * (n/r)^(n + 1/2) / sqrt((1 + 3*r)*exp(3*r) - (1 + 2*r)*exp(2*r)), where r = LambertW(3*n)/3 - 1/(2 + 3/LambertW(3*n) - 3^(4/3) * n^(1/3) * (1 + LambertW(3*n)) / LambertW(3*n)^(4/3)). - Vaclav Kotesovec, Jul 05 2022

A368018 Expansion of e.g.f. exp(exp(2*x) - exp(3*x)).

Original entry on oeis.org

1, -1, -4, -5, 57, 548, 1967, -13561, -302718, -2589819, -2709911, 300801642, 5531279773, 48708116819, -142678610012, -13947271486097, -277586590571059, -2741155101562764, 15789174378252979, 1332483468802350235, 31222229349684528898, 380895661222461566625
Offset: 0

Views

Author

Seiichi Manyama, Dec 08 2023

Keywords

Crossrefs

Cf. A355381.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, (2^j-3^j)*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} (2^k - 3^k) * binomial(n-1,k-1) * a(n-k).
Showing 1-5 of 5 results.

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