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 1-7 of 7 results.

A336950 E.g.f.: 1 / (1 - x * exp(2*x)).

Original entry on oeis.org

1, 1, 6, 42, 392, 4600, 64752, 1063216, 19952256, 421227648, 9880951040, 254960721664, 7176891675648, 218857588139008, 7187394935347200, 252897556424140800, 9491754142468702208, 378509920569294684160, 15982018774576565649408, 712306819507400060502016
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 08 2020

Keywords

Links

Crossrefs

Column k=2 of A351790.
Cf. A001787, A006153, A122704, A216794, A235328, A336947, A336951, A336952.

Programs

  • Mathematica
    nmax = 19; CoefficientList[Series[1/(1 - x Exp[2 x]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[n! Sum[(2 (n - k))^k/k!, {k, 0, n}], {n, 1, 19}]]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] k 2^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]
  • PARI
    seq(n)={ Vec(serlaplace(1 / (1 - x*exp(2*x + O(x^n))))) } \\ Andrew Howroyd, Aug 08 2020

Formula

a(n) = n! * Sum_{k=0..n} (2 * (n-k))^k / k!.
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * k * 2^(k-1) * a(n-k).
a(n) ~ n! * (2/LambertW(2))^n / (1 + LambertW(2)). - Vaclav Kotesovec, Aug 09 2021

A336948 E.g.f.: 1 / (exp(-3*x) - x).

Original entry on oeis.org

1, 4, 23, 195, 2229, 31863, 546255, 10925757, 249753897, 6422808411, 183524701779, 5768419379913, 197791542799965, 7347180526444359, 293912722687075767, 12597352573293062757, 575928946256877156177, 27976119070974574461363, 1438896686251112024068251
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 08 2020

Keywords

Crossrefs

Cf. A072597, A328182, A336947, A336949, A336951.

Programs

  • Mathematica
    nmax = 18; CoefficientList[Series[1/(Exp[-3 x] - x), {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(3 (n - k + 1))^k/k!, {k, 0, n}], {n, 0, 18}]
a[0] = 1; a[n_] := a[n] = 4 n a[n - 1] - Sum[Binomial[n, k] (-3)^k a[n - k], {k, 2, n}]; Table[a[n], {n, 0, 18}]
  • PARI
    seq(n)={ Vec(serlaplace(1 / (exp(-3*x + O(x*x^n)) - x))) } \\ Andrew Howroyd, Aug 08 2020

Formula

a(n) = n! * Sum_{k=0..n} (3 * (n-k+1))^k / k!.
a(0) = 1; a(n) = 4 * n * a(n-1) - Sum_{k=2..n} binomial(n,k) * (-3)^k * a(n-k).
a(n) ~ n! / ((1 + LambertW(3)) * (LambertW(3)/3)^(n+1)). - Vaclav Kotesovec, Aug 09 2021

A336952 E.g.f.: 1 / (1 - x * exp(4*x)).

Original entry on oeis.org

1, 1, 10, 102, 1336, 22200, 443664, 10334128, 275060608, 8236914048, 274069953280, 10031110907136, 400520747437056, 17324601073921024, 807023462798608384, 40278407730378332160, 2144307919689898491904, 121291661335680615284736, 7264376142168665821741056
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 08 2020

Keywords

Links

Crossrefs

Column k=4 of A351790.
Cf. A002697, A006153, A235328, A326324, A328183, A336950, A336951.

Programs

  • Mathematica
    nmax = 18; CoefficientList[Series[1/(1 - x Exp[4 x]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[n! Sum[(4 (n - k))^k/k!, {k, 0, n}], {n, 1, 18}]]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] k 4^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
  • PARI
    seq(n)={ Vec(serlaplace(1 / (1 - x*exp(4*x + O(x^n))))) } \\ Andrew Howroyd, Aug 08 2020

Formula

a(n) = n! * Sum_{k=0..n} (4 * (n-k))^k / k!.
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * k * 4^(k-1) * a(n-k).
a(n) ~ n! * (4/LambertW(4))^n / (1 + LambertW(4)). - Vaclav Kotesovec, Aug 09 2021

A351790 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} (k * (n-j))^j/j!.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 4, 6, 1, 1, 6, 21, 24, 1, 1, 8, 42, 148, 120, 1, 1, 10, 69, 392, 1305, 720, 1, 1, 12, 102, 780, 4600, 13806, 5040, 1, 1, 14, 141, 1336, 11145, 64752, 170401, 40320, 1, 1, 16, 186, 2084, 22200, 191178, 1063216, 2403640, 362880
Offset: 0

Views

Author

Seiichi Manyama, Feb 19 2022

Keywords

Examples

			Square array begins:
    1,    1,    1,     1,     1,     1, ...
    1,    1,    1,     1,     1,     1, ...
    2,    4,    6,     8,    10,    12, ...
    6,   21,   42,    69,   102,   141, ...
   24,  148,  392,   780,  1336,  2084, ...
  120, 1305, 4600, 11145, 22200, 39145, ...

Crossrefs

Columns k=0..4 give A000142, A006153, A336950, A336951, A336952.
Main diagonal gives A235328.
Cf. A351761, A351791.

Programs

  • Mathematica
    T[n_, k_] := n!*(1 + Sum[(k*(n - j))^j/j!, {j, 1, n}]); Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 19 2022 *)
  • PARI
    T(n, k) = n!*sum(j=0, n, (k*(n-j))^j/j!);
  • PARI
    T(n, k) = if(n==0, 1, n*sum(j=0, n-1, k^(n-1-j)*binomial(n-1, j)*T(j, k)));

Formula

E.g.f. of column k: 1/(1 - x*exp(k*x)).
T(0,k) = 1 and T(n,k) = n * Sum_{j=0..n-1} k^(n-1-j) * binomial(n-1,j) * T(j,k) for n > 0.

A351792 Expansion of e.g.f. 1/(1 - x*exp(-3*x)).

Original entry on oeis.org

1, 1, -4, -3, 132, -375, -8298, 86121, 636696, -20318607, 15154290, 5555366289, -57903946092, -1608939709767, 44662076643870, 329040381072825, -31446740971136592, 195779189199531105, 21694625692807192938, -496937940680594097279
Offset: 0

Views

Author

Seiichi Manyama, Feb 19 2022

Keywords

Crossrefs

Column k=3 of A351791.
Cf. A336951, A351778.

Programs

  • Mathematica
    a[0] = 1; a[n_] := n!*Sum[(-3*(n - k))^k/k!, {k, 0, n}]; Array[a, 20, 0] (* Amiram Eldar, Feb 19 2022 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-x*exp(-3*x))))
  • PARI
    a(n) = n!*sum(k=0, n, (-3*(n-k))^k/k!);
  • PARI
    a(n) = if(n==0, 1, n*sum(k=0, n-1, (-3)^(n-1-k)*binomial(n-1, k)*a(k)));

Formula

a(n) = n! * Sum_{k=0..n} (-3 * (n-k))^k/k!.
a(0) = 1 and a(n) = n * Sum_{k=0..n-1} (-3)^(n-1-k) * binomial(n-1,k) * a(k) for n > 0.

A356827 Expansion of e.g.f. exp(x * exp(3*x)).

Original entry on oeis.org

1, 1, 7, 46, 361, 3436, 37729, 463366, 6280369, 93015352, 1491337441, 25684077706, 472217487625, 9221588527204, 190441412508481, 4143470377262806, 94663498086222049, 2264440394856702832, 56570146384760433217, 1472545685988162638722
Offset: 0

Views

Author

Seiichi Manyama, Aug 29 2022

Keywords

Crossrefs

Cf. A000248, A003725, A216689, A295552.
Cf. A277456, A336951, A351737, A355501, A356820.

Programs

  • Maple
    A356827 := proc(n)
    add((3*k)^(n-k) * binomial(n,k),k=0..n) ;
end proc:
seq(A356827(n),n=0..70) ; # R. J. Mathar, Dec 04 2023
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x*exp(3*x))))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-3*k*x)^(k+1)))
  • PARI
    a(n) = sum(k=0, n, (3*k)^(n-k)*binomial(n, k));

Formula

G.f.: Sum_{k>=0} x^k / (1 - 3*k*x)^(k+1).
a(n) = Sum_{k=0..n} (3*k)^(n-k) * binomial(n,k).

A368177 Expansion of e.g.f. -log(1 - x * exp(3*x)).

Original entry on oeis.org

0, 1, 7, 47, 402, 4569, 65298, 1119789, 22397112, 511972065, 13166163630, 376208954109, 11824734538620, 405454640476833, 15061050695642994, 602494304797738845, 25823425094211472272, 1180601869774944168513, 57348495330075309426390
Offset: 0

Views

Author

Seiichi Manyama, Dec 14 2023

Keywords

Crossrefs

Cf. A009306, A009444, A368176.
Cf. A336951.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[-Log[1-x Exp[3x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 10 2024 *)
  • PARI
    a(n) = sum(k=1, n, (3*k)^(n-k)*(k-1)!*binomial(n, k));

Formula

a(n) = Sum_{k=1..n} (3*k)^(n-k) * (k-1)! * binomial(n,k).
a(n) ~ (n-1)! * 3^n / LambertW(3)^n. - Vaclav Kotesovec, Mar 11 2024
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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