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 1-6 of 6 results.

A337038 a(n) = exp(-1/2) * Sum_{k>=0} (2*k - 1)^n / (2^k * k!).

Original entry on oeis.org

1, 0, 2, 4, 20, 96, 552, 3536, 25104, 194816, 1637408, 14792768, 142761280, 1464117760, 15886137984, 181667507456, 2182268117248, 27456279388160, 360872502280704, 4943580063237120, 70437638474568704, 1041911242274562048, 15972832382065977344, 253388070573020401664
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 12 2020

Keywords

Links

Crossrefs

Cf. A000296, A004211, A007405, A124311, A166922, A217203, A337039, A337040, A337041, A337042, A337043.

Programs

  • Maple
    E:= exp((exp(2*x)-1)/2-x):
S:= series(E,x,31):
seq(coeff(S,x,i)*i!,i=0..30); # Robert Israel, Aug 26 2020
  • Mathematica
    nmax = 23; CoefficientList[Series[Exp[(Exp[2 x] - 1)/2 - x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k] 2^k a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 23}]
Table[Sum[(-1)^(n - k) Binomial[n, k] 2^k BellB[k, 1/2], {k, 0, n}], {n, 0, 23}]

Formula

G.f. A(x) satisfies: A(x) = (1 - 2*x + x*A(x/(1 - 2*x))) / (1 - x - 2*x^2).
G.f.: (1/(1 + x)) * Sum_{k>=0} (x/(1 + x))^k / Product_{j=1..k} (1 - 2*j*x/(1 + x)).
E.g.f.: exp((exp(2*x) - 1) / 2 - x).
a(0) = 1; a(n) = Sum_{k=1..n-1} binomial(n-1,k) * 2^k * a(n-k-1).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A004211(k).
a(n) ~ 2^(n - 1/2) * n^(n - 1/2) * exp(n/LambertW(2*n) - n - 1/2) / (sqrt(1 + LambertW(2*n)) * LambertW(2*n)^(n - 1/2)). - Vaclav Kotesovec, Jun 26 2022

A337040 a(n) = exp(-1/4) * Sum_{k>=0} (4*k - 1)^n / (4^k * k!).

Original entry on oeis.org

1, 0, 4, 16, 112, 896, 8384, 88320, 1032448, 13242368, 184591360, 2773929984, 44641579008, 765196926976, 13905753980928, 266855007453184, 5388980396818432, 114172599765827584, 2530858142594760704, 58556990344729198592, 1411095950792925904896, 35347148031264582270976
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 12 2020

Keywords

Crossrefs

Cf. A000296, A003576, A004213, A337038, A337039, A337041, A337042, A337043.

Programs

  • Mathematica
    nmax = 21; CoefficientList[Series[Exp[(Exp[4 x] - 1)/4 - x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k] 4^k a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 21}]
Table[Sum[(-1)^(n - k) Binomial[n, k] 4^k BellB[k, 1/4], {k, 0, n}], {n, 0, 21}]

Formula

G.f. A(x) satisfies: A(x) = (1 - 4*x + x*A(x/(1 - 4*x))) / (1 - 3*x - 4*x^2).
G.f.: (1/(1 + x)) * Sum_{k>=0} (x/(1 + x))^k / Product_{j=1..k} (1 - 4*j*x/(1 + x)).
E.g.f.: exp((exp(4*x) - 1) / 4 - x).
a(0) = 1; a(n) = Sum_{k=1..n-1} binomial(n-1,k) * 4^k * a(n-k-1).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A004213(k).
a(n) ~ 4^(n - 1/4) * n^(n - 1/4) * exp(n/LambertW(4*n) - n - 1/4) / (sqrt(1 + LambertW(4*n)) * LambertW(4*n)^(n - 1/4)). - Vaclav Kotesovec, Jun 26 2022

A337041 a(n) = exp(-1/5) * Sum_{k>=0} (5*k - 1)^n / (5^k * k!).

Original entry on oeis.org

1, 0, 5, 25, 200, 1875, 20625, 256250, 3534375, 53515625, 881468750, 15667578125, 298478828125, 6060493750000, 130542772265625, 2971013486328125, 71193375156250000, 1790666151318359375, 47145509926611328125, 1296156682961425781250, 37129279010879638671875
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 12 2020

Keywords

Crossrefs

Cf. A000296, A003577, A005011, A337038, A337039, A337040, A337042, A337043.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Exp[(Exp[5 x] - 1)/5 - x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k] 5^k a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 20}]
Table[Sum[(-1)^(n - k) Binomial[n, k] 5^k BellB[k, 1/5], {k, 0, n}], {n, 0, 20}]

Formula

G.f. A(x) satisfies: A(x) = (1 - 5*x + x*A(x/(1 - 5*x))) / (1 - 4*x - 5*x^2).
G.f.: (1/(1 + x)) * Sum_{k>=0} (x/(1 + x))^k / Product_{j=1..k} (1 - 5*j*x/(1 + x)).
E.g.f.: exp((exp(5*x) - 1) / 5 - x).
a(0) = 1; a(n) = Sum_{k=1..n-1} binomial(n-1,k) * 5^k * a(n-k-1).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A005011(k).
a(n) ~ 5^(n - 1/5) * n^(n - 1/5) * exp(n/LambertW(5*n) - n - 1/5) / (sqrt(1 + LambertW(5*n)) * LambertW(5*n)^(n - 1/5)). - Vaclav Kotesovec, Jun 26 2022

A337042 a(n) = exp(-1/6) * Sum_{k>=0} (6*k - 1)^n / (6^k * k!).

Original entry on oeis.org

1, 0, 6, 36, 324, 3456, 43416, 618192, 9778320, 169827840, 3210376032, 65540155968, 1435094563392, 33510354739200, 830486180748672, 21756166766173440, 600339119317643520, 17394883290643709952, 527782830161632077312, 16727350847049194775552
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 12 2020

Keywords

Comments

In general, if m >= 1, b <> 0 and e.g.f. = exp(m*exp(b*x) + r*x + s) then a(n) ~ b^n * n^(n + r/b) * exp(n/LambertW(n/m) - n + s) / (m^(r/b) * sqrt(1 + LambertW(n/m)) * LambertW(n/m)^(n + r/b)). - Vaclav Kotesovec, Jun 28 2022

Crossrefs

Cf. A000296, A003578, A005012, A337038, A337039, A337040, A337041, A337043.

Programs

  • Mathematica
    nmax = 19; CoefficientList[Series[Exp[(Exp[6 x] - 1)/6 - x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k] 6^k a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 19}]
Table[Sum[(-1)^(n - k) Binomial[n, k] 6^k BellB[k, 1/6], {k, 0, n}], {n, 0, 19}]

Formula

G.f. A(x) satisfies: A(x) = (1 - 6*x + x*A(x/(1 - 6*x))) / (1 - 5*x - 6*x^2).
G.f.: (1/(1 + x)) * Sum_{k>=0} (x/(1 + x))^k / Product_{j=1..k} (1 - 6*j*x/(1 + x)).
E.g.f.: exp((exp(6*x) - 1) / 6 - x).
a(0) = 1; a(n) = Sum_{k=1..n-1} binomial(n-1,k) * 6^k * a(n-k-1).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A005012(k).
a(n) ~ 6^(n - 1/6) * n^(n - 1/6) * exp(n/LambertW(6*n) - n - 1/6) / (sqrt(1 + LambertW(6*n)) * LambertW(6*n)^(n - 1/6)). - Vaclav Kotesovec, Jun 26 2022

A337043 a(0) = 1; thereafter a(n) = exp(-1/n) * Sum_{k>=0} (n*k - 1)^n / (n^k * k!).

Original entry on oeis.org

1, 0, 2, 9, 112, 1875, 43416, 1310946, 49778688, 2313362673, 128894500000, 8469572721533, 647341071298560, 56871349337125648, 5684260661585401728, 640631299771142578125, 80788871646072851660800, 11323828537291632967145015, 1753760620207362607774290432
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 12 2020

Keywords

Crossrefs

Cf. A000296, A301419, A330605, A334162, A337038, A337039, A337040, A337041, A337042.

Programs

  • Mathematica
    Table[SeriesCoefficient[1/(1 + x) Sum[(x/(1 + x))^k/Product[(1 - n j x/(1 + x)), {j, 1, k}], {k, 0, n}], {x, 0, n}], {n, 0, 18}]
Join[{1}, Table[n! SeriesCoefficient[Exp[(Exp[n x] - 1)/n - x], {x, 0, n}], {n, 1, 18}]]
Join[{1}, Table[Sum[(-1)^(n - k) Binomial[n,k] n^k BellB[k, 1/n], {k, 0, n}], {n, 1, 18}]]

Formula

a(n) = [x^n] (1/(1 + x)) * Sum_{k>=0} (x/(1 + x))^k / Product_{j=1..k} (1 - n*j*x/(1 + x)).
a(n) = n! * [x^n] exp((exp(n*x) - 1) / n - x), for n > 0.
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * n^k * BellPolynomial_k(1/n), for n > 0.

A367886 Expansion of e.g.f. 1/(1 - x * (exp(3*x) - 1)).

Original entry on oeis.org

1, 0, 6, 27, 324, 3645, 54918, 923643, 18061704, 394663833, 9607469130, 256997250279, 7502660832780, 237243300445125, 8079508278302958, 294800526215739315, 11473728720705019152, 474469344621574172721, 20774758472643152149650
Offset: 0

Views

Author

Seiichi Manyama, Dec 04 2023

Keywords

Crossrefs

Cf. A052848, A367885.
Cf. A337039, A351737.

Programs

  • PARI
    a(n) = n!*sum(k=0, n\2, 3^(n-k)*k!*stirling(n-k, k, 2)/(n-k)!);

Formula

a(0) = 1; a(n) = n * Sum_{k=2..n} 3^(k-1) * binomial(n-1,k-1) * a(n-k).
a(n) = n! * Sum_{k=0..floor(n/2)} 3^(n-k) * k! * Stirling2(n-k,k)/(n-k)!.
Showing 1-6 of 6 results.

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