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.

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.

A367786 Expansion of e.g.f. exp(exp(4*x) - x - 1).

Original entry on oeis.org

1, 3, 25, 235, 2737, 36947, 563657, 9542715, 176920417, 3555369635, 76820077945, 1772943290763, 43469116126737, 1127040956393203, 30779951676185385, 882453651485815003, 26480355971228530369, 829522636694530362691, 27064267045022876869337, 917751849133986186857003
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 30 2023

Keywords

Crossrefs

Cf. A000110, A000296, A124311, A285064, A285065, A330605, A367785.

Programs

  • Mathematica
    nmax = 19; CoefficientList[Series[Exp[Exp[4 x] - x - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -a[n - 1] + Sum[Binomial[n - 1, k - 1] 4^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]
Table[Sum[(-1)^(n - k) Binomial[n, k] 4^k BellB[k], {k, 0, n}], {n, 0, 19}]
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(exp(exp(4*x) - x - 1))) \\ Michel Marcus, Nov 30 2023

Formula

a(n) = exp(-1) * Sum_{k>=0} (4*k-1)^n / k!.
a(0) = 1; a(n) = -a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * 4^k * a(n-k).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * 4^k * Bell(k).

A367785 Expansion of e.g.f. exp(exp(3*x) - x - 1).

Original entry on oeis.org

1, 2, 13, 89, 772, 7745, 87949, 1109288, 15332539, 229840361, 3706130914, 63857565095, 1169261937973, 22646779177898, 462143532144937, 9902312863237637, 222119823632283628, 5202170552214520637, 126914730275907871201, 3218552632981994910248, 84686139239808135094879, 2307953474037054591248501
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 30 2023

Keywords

Crossrefs

Cf. A000110, A000296, A124311, A247452, A284859, A284860, A330605, A367744, A367786.

Programs

  • Mathematica
    nmax = 21; CoefficientList[Series[Exp[Exp[3 x] - x - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -a[n - 1] + Sum[Binomial[n - 1, k - 1] 3^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]
Table[Sum[(-1)^(n - k) Binomial[n, k] 3^k BellB[k], {k, 0, n}], {n, 0, 21}]
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(exp(exp(3*x) - x - 1))) \\ Michel Marcus, Nov 30 2023

Formula

a(n) = exp(-1) * Sum_{k>=0} (3*k-1)^n / k!.
a(0) = 1; a(n) = -a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * 3^k * a(n-k).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * 3^k * Bell(k).

A330604 a(n) = Sum_{k>=0} (n*k - 1)^n / 2^(k + 1).

Original entry on oeis.org

1, 0, 9, 278, 16145, 1471774, 194652577, 35275961958, 8397548586177, 2542220603893358, 954003495852753401, 434683708245705663766, 236409592518584290327249, 151286889086525353482149022, 112534788142976814403622739921, 96285847680519841273313314779974
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 19 2019

Keywords

Crossrefs

Cf. A000670, A052841, A308864, A330605.

Programs

  • Mathematica
    Table[Sum[(n k - 1)^n/2^(k + 1), {k, 0, Infinity}], {n, 0, 15}]
Join[{1}, Table[n^n HurwitzLerchPhi[1/2, -n, -1/n]/2, {n, 1, 15}]]
Table[n! SeriesCoefficient[Exp[-x]/(2 - Exp[n x]), {x, 0, n}], {n, 0, 15}]

Formula

a(n) = n! * [x^n] exp(-x) / (2 - exp(n*x)).
a(n) = Sum_{k=0..n} (-1)^(n - k) * binomial(n,k) * n^k * A000670(k).
a(n) ~ n^n * n! / (2 * (log(2))^(n+1)). - Vaclav Kotesovec, Dec 19 2019

A367784 a(n) = exp(1) * Sum_{k>=0} (-1)^k * (n*k - 1)^n / k!.

Original entry on oeis.org

1, -2, 5, 17, 17, -8151, -311435, -777974, 927723585, 82906687673, 1693962380101, -707005824990631, -137258747025993071, -10253960705018807830, 1697644859939460151413, 803696888217607331079149, 148126297324647875348070657, -323461353221296480463456191
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 30 2023

Keywords

Crossrefs

Cf. A000587, A109747, A307066, A307080, A330605, A367743, A367744.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[Exp[1 - x - Exp[n x]], {x, 0, n}], {n, 0, 17}]
Unprotect[Power]; 0^0 = 1; Table[Sum[(-1)^(n - k) Binomial[n, k] n^k BellB[k, -1], {k, 0, n}], {n, 0, 17}]

Formula

a(n) = n! * [x^n] exp(1 - x - exp(n*x)).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * n^k * A000587(k).
Showing 1-5 of 5 results.

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