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.

Previous Showing 11-18 of 18 results.

A328054 Expansion of e.g.f. log(1 + x / (1 - x)^2).

Original entry on oeis.org

0, 1, 3, 8, 18, 24, 0, 720, 15120, 161280, 1088640, 3628800, 0, 479001600, 18681062400, 348713164800, 3923023104000, 20922789888000, 0, 6402373705728000, 364935301226496000, 9731608032706560000, 153272826515128320000, 1124000727777607680000, 0, 620448401733239439360000
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 03 2019

Keywords

Comments

Logarithmic transform of A001563.

Links

Crossrefs

Cf. A001563, A008588 (positions of 0's), A009306, A082579, A328055.

Programs

  • Maple
    b:= proc(n) option remember; n*n! end:
a:= proc(n) option remember; `if`(n=0, 0, b(n)-
      add(binomial(n, j)*j*b(n-j)*a(j), j=1..n-1)/n)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 04 2019
  • Mathematica
    nmax = 25; CoefficientList[Series[Log[1 + x/(1 - x)^2], {x, 0, nmax}], x] Range[0, nmax]!
Join[{0}, Table[2 (n - 1)! (1 - Cos[Pi n/3]), {n, 1, 25}]]
  • PARI
    my(x='x+O('x^30)); concat(0, Vec(serlaplace(log(1 + x / (1 - x)^2)))) \\ Michel Marcus, Oct 04 2019

Formula

E.g.f.: log(1 + Sum_{k>=1} k * x^k).
D-finite with recurrence a(n+3) = n*(n+1)*(n+2)*a(n) - 2*(n+2)*(n+1)*a(n+1) + 2*(n+2)*a(n+2). - Robert Israel, Jan 16 2023

A337590 a(0) = 0; a(n) = n - (1/n) * Sum_{k=1..n-1} binomial(n,k)^2 * (n-k) * k * a(k).

Original entry on oeis.org

0, 1, 0, -3, 28, -215, -174, 90223, -3840472, 103719537, 429704110, -357346077869, 35100093531900, -2005608652057595, -24108041118593418, 27881407632242902515, -4876442148527153942384, 474102062424164433715937, 12637408141631813073125094, -18867461801192524662360616421
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 02 2020

Keywords

Crossrefs

Cf. A002190, A009306, A336227.

Programs

  • Mathematica
    a[0] = 0; a[n_] := a[n] = n - (1/n) Sum[Binomial[n, k]^2 (n - k) k a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 19}]
nmax = 19; CoefficientList[Series[Log[1 + Sqrt[x] BesselI[1, 2 Sqrt[x]]], {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

Sum_{n>=0} a(n) * x^n / (n!)^2 = log(1 + sqrt(x) * BesselI(1,2*sqrt(x))).
Sum_{n>=0} a(n) * x^n / (n!)^2 = log(1 + Sum_{n>=1} n * x^n / (n!)^2).

A307126 Expansion of e.g.f. log(1 + log(1 + x*exp(x))).

Original entry on oeis.org

0, 1, 0, -2, 5, 3, -88, 362, 534, -17363, 103354, 175690, -9218328, 80446715, 46936658, -10553663682, 136009808336, -210505566343, -22766371152222, 418488315816586, -1679396876267976, -82907733267235305, 2070045795782097506, -13611715282931011890, -463120892871268874832
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 26 2019

Keywords

Crossrefs

Cf. A003713, A007550, A009306, A307125.

Programs

  • Maple
    a:=series(log(1+log(1+x*exp(x))),x=0,25): seq(n!*coeff(a,x,n),n=0..24); # Paolo P. Lava, Apr 03 2019
  • Mathematica
    nmax = 24; CoefficientList[Series[Log[1 + Log[1 + x Exp[x]]], {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    my(x = 'x + O('x^30)); Vec(serlaplace(log(1 + log(1 + x*exp(x))))) \\ Michel Marcus, Mar 26 2019

A308484 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. log(1 + Sum_{j>=1} j^k * x^j/j!).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, -1, 0, 1, 7, -1, -2, 0, 1, 15, 5, -26, 9, 0, 1, 31, 35, -146, 29, 6, 0, 1, 63, 149, -650, -351, 756, -155, 0, 1, 127, 539, -2642, -5251, 9936, -1793, 232, 0, 1, 255, 1805, -10346, -46071, 83376, 51421, -45744, 3969, 0
Offset: 1

Views

Author

Seiichi Manyama, May 30 2019

Keywords

Examples

			Square array begins:
   1,    1,     1,     1,       1,        1, ...
   0,    1,     3,     7,      15,       31, ...
   0,   -1,    -1,     5,      35,      149, ...
   0,   -2,   -26,  -146,    -650,    -2642, ...
   0,    9,    29,  -351,   -5251,   -46071, ...
   0,    6,   756,  9936,   83376,   559656, ...
   0, -155, -1793, 51421, 1623439, 28735405, ...

Links

Crossrefs

Columns k=0..4 give A000007(n-1), A009306, A033464, A300452, A306325.
A(n,n) gives A320939.

Programs

  • Mathematica
    T[n_, k_] := T[n, k] = n^k - Sum[Binomial[n-1,j] * j^k * T[n-j,k], {j,1,n-1}]; Table[T[k, n - k], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, May 12 2021 *)

Formula

A(n,k) = n^k - Sum_{j=1..n-1} binomial(n-1,j)*j^k*A(n-j,k).

A320255 a(n) = n! * [x^n] log(1 + exp(x)*(x + (n/2 - 1)*x^2)).

Original entry on oeis.org

0, 1, 1, -1, -26, 39, 3666, -7400, -1488416, 3802113, 1322570530, -4095154284, -2187371499312, 7964242253473, 6052757424558586, -25343867475914910, -25988018018090461664, 123032891453320498449, 163684285184147641156098, -864557405968781387651984, -1448111703094244548802632160
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 08 2018

Keywords

Comments

For n > 2, a(n) is the n-th term of the logarithmic transform of n-gonal numbers.

Links

Crossrefs

Cf. A009306, A033464, A300455, A318118, A318124, A320254.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[Log[1 + Exp[x] (x + (n/2 - 1) x^2)], {x, 0, n}], {n, 0, 20}]

A368176 Expansion of e.g.f. -log(1 - x * exp(2*x)).

Original entry on oeis.org

0, 1, 5, 26, 182, 1704, 19992, 281392, 4620464, 86707584, 1830550400, 42940149504, 1107995749632, 31188982438912, 951100528802816, 31234626965637120, 1099029746752575488, 41248797730190032896, 1644909773059509682176
Offset: 0

Views

Author

Seiichi Manyama, Dec 14 2023

Keywords

Crossrefs

Cf. A009306, A009444, A368177.
Cf. A336950.

Programs

  • PARI
    a(n) = sum(k=1, n, (2*k)^(n-k)*(k-1)!*binomial(n, k));

Formula

a(n) = Sum_{k=1..n} (2*k)^(n-k) * (k-1)! * binomial(n,k).
a(n) ~ (n-1)! * 2^n / LambertW(2)^n. - Vaclav Kotesovec, Mar 11 2024

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

A344469 Triangle read by rows: T(n, k) (0 <= k <= n) = [x^k] x^n * n! * [t^n] x*(1 + t)/(x*exp(-t) - t).

Original entry on oeis.org

1, 1, 2, 2, 6, 3, 6, 24, 24, 4, 24, 120, 180, 80, 5, 120, 720, 1440, 1080, 240, 6, 720, 5040, 12600, 13440, 5670, 672, 7, 5040, 40320, 120960, 168000, 107520, 27216, 1792, 8, 40320, 362880, 1270080, 2177280, 1890000, 774144, 122472, 4608, 9
Offset: 0

Views

Author

Peter Luschny, May 20 2021

Keywords

Comments

Related to the Lambert W-function, see Cohen, Corollary 2.4.

Examples

			Triangle starts:
[0] 1;
[1] 1,     2;
[2] 2,     6,      3;
[3] 6,     24,     24,      4;
[4] 24,    120,    180,     80,      5;
[5] 120,   720,    1440,    1080,    240,     6;
[6] 720,   5040,   12600,   13440,   5670,    672,    7;
[7] 5040,  40320,  120960,  168000,  107520,  27216,  1792,   8;
[8] 40320, 362880, 1270080, 2177280, 1890000, 774144, 122472, 4608, 9.

Links

Crossrefs

Cf. A305990 (row sums), A009306 (alternating row sums).

Programs

  • Maple
    gf := x*(1+t)/(x*exp(-t)-t): ser := series(gf,t,12):
seq(seq(coeff(expand(x^n*n!*coeff(ser,t,n)),x,k),k=0..n),n=0..8);
  • Mathematica
    (* rows[n], n[0..oo] *)
n=12;r={};For[k=0,kDetlef Meya, Jul 31 2023 *)
Previous Showing 11-18 of 18 results.

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