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.

A355381 Expansion of e.g.f. exp(exp(3*x) - exp(2*x)).

Original entry on oeis.org

1, 1, 6, 35, 247, 2102, 20547, 224541, 2707292, 35638329, 507464939, 7757439428, 126538995293, 2191454313661, 40120212534838, 773554002955047, 15656660861190371, 331700076893737054, 7337160433117899959, 169068422994937678185, 4050093664805130165348
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 30 2022

Keywords

Comments

In general, if m > 0, b > d >= 1 and e.g.f. = exp(m*exp(b*x) + r*exp(d*x) + s) then a(n) ~ exp(m*exp(b*z) + r*exp(d*z) + s - n) * (n/z)^(n + 1/2) / sqrt(m*b*(1 + b*z)*exp(b*z) + r*d*(1 + d*z)*exp(d*z)), where z = LambertW(n/m)/b - 1/(d + b/LambertW(n/m) + b^2 * m^(d/b) * n^(1 - d/b) * (1 + LambertW(n/m)) / (d*r*LambertW(n/m)^(2 - d/b))). - Vaclav Kotesovec, Jul 03 2022
In addition, if b/d >=2 then a(n) ~ c * (b*n/LambertW(n/m))^n * exp(n/LambertW(n/m) + r * (n/(m*LambertW(n/m)))^(d/b) - n + s) / sqrt(1 + LambertW(n/m)), where c = 1 for b/d > 2 and c = exp(-r^2/(8*m)) for b/d = 2. - Vaclav Kotesovec, Jul 10 2022

Links

Crossrefs

Cf. A143405, A194006, A355291, A355380, A355378.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Exp[Exp[3*x] - Exp[2*x]], {x, 0, nmax}], x] * Range[0, nmax]!
Table[Sum[Binomial[n,k] * 3^k * 2^(n-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(2*x)))) \\ 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, -1).
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) - 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

A355408 Expansion of e.g.f. 1/(1 + exp(x) - exp(3*x)).

Original entry on oeis.org

1, 2, 16, 170, 2416, 42962, 916696, 22819610, 649207456, 20778364322, 738918769576, 28905116527850, 1233506128752496, 57025618592932082, 2839117599033828856, 151446758367400488890, 8617182795217834505536, 520954229292164353554242
Offset: 0

Views

Author

Seiichi Manyama, Jul 01 2022

Keywords

Crossrefs

Cf. A000556, A355378, A355409.

Programs

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

Formula

a(0) = 1; a(n) = Sum_{k=1..n} (3^k - 1) * binomial(n,k) * a(n-k).
a(n) ~ n! / ((3 + 2*r) * log(r)^(n+1)), where r = 2*cosh(log((25 + 3*sqrt(69)) / 2) / 6)/sqrt(3). - Vaclav Kotesovec, Jul 01 2022

A355379 Expansion of e.g.f. exp(exp(3*x) + exp(x) - 2).

Original entry on oeis.org

1, 4, 26, 212, 2046, 22588, 278942, 3792916, 56128254, 895795692, 15307847614, 278435732484, 5364073445278, 108994074306268, 2327475127169182, 52069279762495220, 1217024509006768574, 29647115491635327180, 751085909757123127294, 19750410883486281805028
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 30 2022

Keywords

Links

Crossrefs

Cf. A143405, A355291, A355378.

Programs

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

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * 3^k * Bell(k) * Bell(n-k).
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) - 2 - 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 - 2) / sqrt(1 + LambertW(n)). - Vaclav Kotesovec, Jul 10 2022

A368017 Expansion of e.g.f. exp(exp(x) - exp(3*x)).

Original entry on oeis.org

1, -2, -4, 14, 144, 286, -5080, -61058, -186144, 4016958, 73395928, 468915102, -4728823088, -167453193314, -2051810224568, -406640603074, 533831885402048, 11987797433266302, 110763307665075640, -1459040819952150178, -80503810962755821904
Offset: 0

Views

Author

Seiichi Manyama, Dec 08 2023

Keywords

Crossrefs

Cf. A355378.

Programs

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

Formula

a(0) = 1; a(n) = Sum_{k=1..n} (1 - 3^k) * binomial(n-1,k-1) * a(n-k).

A355396 Expansion of e.g.f. exp(exp(3*x)/3 - exp(x) + 2/3).

Original entry on oeis.org

1, 0, 2, 8, 38, 240, 1782, 14728, 134598, 1352800, 14800502, 174593848, 2205456838, 29676417680, 423455081142, 6381678299368, 101217742764358, 1684357485887680, 29328589792496502, 533062885681064088, 10091434399407455558, 198592474864415055600
Offset: 0

Views

Author

Seiichi Manyama, Jun 30 2022

Keywords

Crossrefs

Cf. A002874, A355378.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Exp[Exp[3*x]/3 - Exp[x] + 2/3], {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(x)+2/3)))
  • 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)-1)*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} (3^(k-1) - 1) * binomial(n-1,k-1) * a(n-k).
a(n) ~ exp(exp(3*r)/3 - exp(r) + 2/3 - n) * (n/r)^(n + 1/2) / sqrt((1 + 3*r)*exp(3*r) - (1 + r)*exp(r)), where r = LambertW(3*n)/3 - 1/(1 + 3/LambertW(3*n) - 3^(5/3) * n^(2/3) * (1 + LambertW(3*n)) / LambertW(3*n)^(5/3)). - Vaclav Kotesovec, Jul 05 2022
a(n) ~ (3*n/LambertW(3*n))^n * exp(n/LambertW(3*n) - (n/(LambertW(3*n)/3))^(1/3) - n + 2/3) / sqrt(1 + LambertW(3*n)). - Vaclav Kotesovec, Jul 10 2022
Showing 1-5 of 5 results.

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