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 1-4 of 4 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

A355421 Expansion of e.g.f. exp(Sum_{k=1..3} (exp(k*x) - 1)).

Original entry on oeis.org

1, 6, 50, 504, 5870, 76872, 1111646, 17522664, 298133054, 5433157512, 105396184478, 2165189912040, 46901678992958, 1067332196912136, 25435754924426270, 633014456504059368, 16411191933603611198, 442258823578968351624
Offset: 0

Views

Author

Seiichi Manyama, Jul 01 2022

Keywords

Crossrefs

Column k=3 of A355423.
Cf. A004701, A306027, A355379, A355380.

Programs

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

Formula

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

A355397 Expansion of e.g.f. exp(exp(3*x)/3 + exp(2*x)/2 - 5/6).

Original entry on oeis.org

1, 2, 9, 51, 350, 2799, 25373, 255854, 2831177, 34023919, 440414146, 6099346455, 89873849705, 1402403637418, 23081230257449, 399284248276827, 7238080522101270, 137125745341692863, 2708536196071195365, 55660194042713099510, 1187724805063462045289
Offset: 0

Views

Author

Seiichi Manyama, Jun 30 2022

Keywords

Crossrefs

Cf. A355380.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Exp[Exp[3*x]/3 + Exp[2*x]/2 - 5/6], {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)/2-5/6)))
  • 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 - 5/6 - 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

A355411 Expansion of e.g.f. 1/(3 - exp(2*x) - exp(3*x)).

Original entry on oeis.org

1, 5, 63, 1175, 29211, 907775, 33852603, 1472830175, 73232729451, 4096474833695, 254608472798043, 17407167078420575, 1298290575826434891, 104900562662494154015, 9127848307446874753083, 850985644429074730049375, 84626187772620135685119531
Offset: 0

Views

Author

Seiichi Manyama, Jul 01 2022

Keywords

Crossrefs

Cf. A355380, A355409.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(3-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! / ((9 - r^2) * log(r)^(n+1)), where r = (-1 + 2*cosh(log((79 + 9*sqrt(77))/2)/3))/3. - Vaclav Kotesovec, Jul 01 2022
Showing 1-4 of 4 results.

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