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 1-8 of 8 results.

A009444 E.g.f. log(1 + x*exp(-x)).

Original entry on oeis.org

0, 1, -3, 11, -58, 409, -3606, 38149, -470856, 6641793, -105398650, 1858413061, -36044759796, 762659322385, -17481598316742, 431535346662645, -11413394655983536, 321989729198400385, -9651573930139850610
Offset: 0

Views

Author

R. H. Hardin

Keywords

Comments

abs(a(n)) is the number of connected functions f:{1,2,...,n}->{1,2,...,n} such that every element is mapped into a recurrent element. Cf. A006153. - Geoffrey Critzer, May 24 2012

Links

Crossrefs

Cf. A006153, A009306, A305990.

Programs

  • Mathematica
    With[{nmax = 40}, CoefficientList[Series[Log[1 + x*Exp[-x]], {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Nov 22 2017 *)
  • Maxima
    a(n):=(-1)^(n+1)*n!*sum(m^(n-m-1)/(n-m)!,m,1,n); /* Vladimir Kruchinin, Oct 08 2011 */
  • PARI
    x='x+O('x^66); /* that many terms */
egf=1/(1+x/exp(x)); /* = 1 - x + 2*x^2 - 7/2*x^3 + 37/6*x^4 - 87/8*x^5 +... */
Vec(serlaplace(egf)) /* show terms */ /* Joerg Arndt, Apr 30 2011 */
  • Sage
    A009444 = lambda n: (-1)^(n+1)*factorial(n)*sum(m^(n-m-1)/factorial(n-m) for m in (1..n))
[A009444(n) for n in (0..9)] # Peter Luschny, Jan 18 2016

Formula

abs(a(n)) is asymptotic to (n-1)!/LambertW(1)^n. - Vladeta Jovovic, Jul 12 2007
Sequence of absolute values has e.g.f. log(1/(1-x*exp(x))). - Joerg Arndt, Apr 30 2011
a(n) = (-1)^(n+1)*n!*sum(m=1..n, m^(n-m-1)/(n-m)!). - Vladimir Kruchinin, Oct 08 2011
a(n) = (-1)^(n + 1) * n + Sum_{k=1..n-1} (-1)^(n - k) * binomial(n-1,k-1) * (n - k) * a(k). - Ilya Gutkovskiy, Jan 17 2020

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997
Definition corrected by Joerg Arndt, Apr 30 2011

A346753 Expansion of e.g.f. -log( 1 - x^2 * exp(x) / 2 ).

Original entry on oeis.org

0, 0, 1, 3, 9, 40, 225, 1491, 11578, 102852, 1026945, 11394955, 139091106, 1852061718, 26716291693, 415033647315, 6908006807640, 122645325067576, 2313546734841633, 46209268921868595, 974228913850588750, 21620679147700290210, 503810188866302511501
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 01 2021

Keywords

Crossrefs

Cf. A000217, A009444, A133189, A292953, A305990, A346750, A346754, A346755.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[-Log[1 - x^2 Exp[x]/2], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 0; a[n_] := a[n] = Binomial[n, 2] + (1/n) Sum[Binomial[n, k] Binomial[n - k, 2] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 22}]

Formula

a(0) = 0; a(n) = binomial(n,2) + (1/n) * Sum_{k=1..n-1} binomial(n,k) * binomial(n-k,2) * k * a(k).
a(n) ~ (n-1)! / (2*LambertW(1/sqrt(2)))^n. - Vaclav Kotesovec, Aug 08 2021
a(n) = n! * Sum_{k=1..floor(n/2)} k^(n-2*k-1)/(2^k * (n-2*k)!). - Seiichi Manyama, Dec 14 2023

A346754 Expansion of e.g.f. -log( 1 - x^3 * exp(x) / 3! ).

Original entry on oeis.org

0, 0, 0, 1, 4, 10, 30, 175, 1176, 7364, 50520, 425205, 4010380, 39433966, 414654604, 4793188855, 59834495280, 789420239560, 11016095913456, 163423065359529, 2565467553034740, 42320595474149650, 732058678770177220, 13275485607004016011
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 01 2021

Keywords

Crossrefs

Cf. A000292, A009444, A145453, A292954, A305990, A346751, A346753, A346755.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[-Log[1 - x^3 Exp[x]/3!], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 0; a[n_] := a[n] = Binomial[n, 3] + (1/n) Sum[Binomial[n, k] Binomial[n - k, 3] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 23}]

Formula

a(0) = 0; a(n) = binomial(n,3) + (1/n) * Sum_{k=1..n-1} binomial(n,k) * binomial(n-k,3) * k * a(k).
a(n) ~ (n-1)! / (3*LambertW(2^(1/3)/3^(2/3)))^n. - Vaclav Kotesovec, Aug 08 2021
a(n) = n! * Sum_{k=1..floor(n/3)} k^(n-3*k-1)/(6^k * (n-3*k)!). - Seiichi Manyama, Dec 14 2023

A346755 Expansion of e.g.f. -log( 1 - x^4 * exp(x) / 4! ).

Original entry on oeis.org

0, 0, 0, 0, 1, 5, 15, 35, 105, 756, 6510, 46530, 289245, 1892605, 16187171, 170721915, 1833783770, 18875258780, 196470797580, 2255939795436, 29179692064545, 401813199660285, 5612352516200815, 79620308330422475, 1182881543312932386
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 01 2021

Keywords

Crossrefs

Cf. A000332, A009444, A145454, A292955, A305990, A346752, A346753, A346754.

Programs

  • Mathematica
    nmax = 24; CoefficientList[Series[-Log[1 - x^4 Exp[x]/4!], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 0; a[n_] := a[n] = Binomial[n, 4] + (1/n) Sum[Binomial[n, k] Binomial[n - k, 4] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 24}]

Formula

a(0) = 0; a(n) = binomial(n,4) + (1/n) * Sum_{k=1..n-1} binomial(n,k) * binomial(n-k,4) * k * a(k).
a(n) ~ (n-1)! / (4*LambertW(3^(1/4)/2^(5/4)))^n. - Vaclav Kotesovec, Aug 08 2021
a(n) = n! * Sum_{k=1..floor(n/4)} k^(n-4*k-1)/(24^k * (n-4*k)!). - Seiichi Manyama, Dec 14 2023

A305133 E.g.f.: (1-x) / (exp(-x) - x).

Original entry on oeis.org

1, 1, 3, 16, 113, 996, 10537, 130054, 1834513, 29111896, 513307601, 9955832514, 210652214665, 4828548335092, 119193293536969, 3152465052989326, 88935973854834593, 2665836978234855984, 84608363388300429601, 2834484567764492239354, 99956558270008377397081, 3701159405682998540166796, 143571313108884280622221913, 5822409005523822986360056326
Offset: 0

Views

Author

Paul D. Hanna, Jun 15 2018

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 113*x^4/4! + 996*x^5/5! + 10537*x^6/6! + 130054*x^7/7! + 1834513*x^8/8! + 29111896*x^9/9! + ...
RELATED TABLE.
The table of coefficients of x^k in exp(n*x) * A(x) begins:
n=0: [1, (1), 3/2, 8/3, 113/24, 83/10, 10537/720, 65027/2520, ...];
n=1: [(1), 2, (3), 29/6, 25/3, 1757/120, 929/36, 45863/1008, ...];
n=2: [1, (3), 11/2, (9), 361/24, 1559/60, 729/16, 101107/1260, ...];
n=3: [1, 4, (9), 97/6, (82/3), 1863/40, 3637/45, 714319/5040, ...];
n=4: [1, 5, 27/2, (82/3), 1169/24, (251/3), 103801/720, 632897/2520, ...];
n=5: [1, 6, 19, 87/2, (251/3), 17821/120, (5147/20), 2250499/5040, ...];
n=6: [1, 7, 51/2, 197/3, 3305/24, (5147/20), 65633/144, (14293/18), ...];
n=7: [1, 8, 33, 569/6, 652/3, 51893/120, (14293/18), 7078303/5040, ...]; ...
in which terms along the diagonals (enclosed in parenthesis) are equal:
[x^n] exp((n+1)*x) * A(x) = [x^(n+1)] exp(n*x) * A(x) for n >= 0.

Crossrefs

Cf. A072597, A305990.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[(1-x)/(Exp[-x]-x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 15 2022 *)
  • PARI
    {a(n) = n!*polcoeff( (1-x) / (exp(-x +x*O(x^n)) - x), n)}
for(n=0,30,print1(a(n),", "))

Formula

E.g.f. A(x) satisfies: [x^n] exp((n+1)*x) * A(x) = [x^(n+1)] exp(n*x) * A(x) for n >= 0.
a(n) ~ n! * (1 - LambertW(1)) / ((1 + LambertW(1)) * LambertW(1)^(n+1)). - Vaclav Kotesovec, Jun 16 2018
a(n) = 1 + n * Sum_{k=1..n-1} binomial(n-1,k) * a(k). - Ilya Gutkovskiy, Aug 08 2020

A336183 a(n) = n^2 + (1/n) * Sum_{k=1..n-1} binomial(n,k) * k * a(k) * (n-k)^2.

Original entry on oeis.org

1, 5, 23, 154, 1389, 15636, 211231, 3329264, 59969097, 1215233380, 27362096211, 677690995488, 18310602210445, 535964033279780, 16894811428737495, 570603293774677696, 20556251540382371217, 786832900592755991364, 31889277719673937849243, 1364231113649221829763200
Offset: 1

Views

Author

Ilya Gutkovskiy, Jul 10 2020

Keywords

Crossrefs

Cf. A000290, A033462, A033464, A305990, A308861, A336184.

Programs

  • Mathematica
    a[n_] := a[n] = n^2 + (1/n) Sum[Binomial[n, k] k a[k] (n - k)^2, {k, 1, n - 1}]; Table[a[n], {n, 1, 20}]
nmax = 20; CoefficientList[Series[-Log[1 - Exp[x] x (1 + x)], {x, 0, nmax}], x] Range[0, nmax]! // Rest

Formula

E.g.f.: -log(1 - exp(x) * x * (1 + x)).
E.g.f.: -log(1 - Sum_{k>=1} k^2 * x^k / k!).
a(n) ~ (n-1)! / r^n, where r = A201941 = 0.444130228823966590585466329490984667... is the root of the equation exp(r)*r*(1+r) = 1. - Vaclav Kotesovec, Jul 11 2020

A336184 a(n) = n^3 + (1/n) * Sum_{k=1..n-1} binomial(n,k) * k * a(k) * (n-k)^3.

Original entry on oeis.org

1, 9, 53, 466, 5569, 82656, 1474045, 30664656, 729036801, 19499288680, 579487528861, 18943592776032, 675568129695601, 26099852672860344, 1085904530481561645, 48407032164910589056, 2301727955153266523521, 116286277045753464506568, 6220517619913795356269725
Offset: 1

Views

Author

Ilya Gutkovskiy, Jul 10 2020

Keywords

Crossrefs

Cf. A000578, A279358, A300452, A305990, A308862, A336183.

Programs

  • Mathematica
    a[n_] := a[n] = n^3 + (1/n) Sum[Binomial[n, k] k a[k] (n - k)^3, {k, 1, n - 1}]; Table[a[n], {n, 1, 19}]
nmax = 19; CoefficientList[Series[-Log[1 - Exp[x] x (1 + 3 x + x^2)], {x, 0, nmax}], x] Range[0, nmax]! // Rest

Formula

E.g.f.: -log(1 - exp(x) * x * (1 + 3*x + x^2)).
E.g.f.: -log(1 - Sum_{k>=1} k^3 * x^k / k!).
a(n) ~ (n-1)! / r^n, where r = 0.336491770414014560614859141224061461582454518... is the root of the equation exp(r)*r*(1 + 3*r + r^2) = 1. - Vaclav Kotesovec, Jul 11 2020

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 *)
Showing 1-8 of 8 results.

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