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 21-27 of 27 results.

A332257 E.g.f.: (1 - sinh(x)) / (1 - 2*sinh(x)).

Original entry on oeis.org

1, 1, 4, 25, 208, 2161, 26944, 391945, 6515968, 121866721, 2532496384, 57890223865, 1443611004928, 38999338931281, 1134616226381824, 35367467110007785, 1175946733416153088, 41543231955279099841, 1553948045857778827264, 61355543097139813855705
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 08 2020

Keywords

Links

Crossrefs

Cf. A000557, A000670, A002866, A006154, A050351, A331608.

Programs

  • Mathematica
    nmax = 19; CoefficientList[Series[(1 - Sinh[x])/(1 - 2 Sinh[x]), {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    seq(n)={Vec(serlaplace((1 - sinh(x + O(x*x^n))) / (1 - 2*sinh(x + O(x*x^n)))))} \\ Andrew Howroyd, Feb 08 2020

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * A006154(k) * a(n-k).
a(n) ~ n! / (2*sqrt(5) * log((1 + sqrt(5))/2)^(n+1)). - Vaclav Kotesovec, Feb 08 2020

A308877 Expansion of e.g.f. (1 + log(1 - x))/(1 + 2*log(1 - x)).

Original entry on oeis.org

1, 1, 5, 38, 386, 4904, 74776, 1330272, 27046848, 618653280, 15723024864, 439559609664, 13405656582336, 442915145716224, 15759326934391296, 600783539885546496, 24430204949876794368, 1055516761826050203648, 48286612866726631489536, 2331682676308057000255488
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 29 2019

Keywords

Crossrefs

Cf. A001710, A002866, A008275, A011782, A050351, A088500, A320349, A308878.

Programs

  • Mathematica
    nmax = 19; CoefficientList[Series[(1 + Log[1 - x])/(1 + 2 Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[Sum[Abs[StirlingS1[n, k]] 2^(k - 1) k!, {k, 1, n}], {n, 1, 19}]]

Formula

a(0) = 1; a(n) = Sum_{k=1..n} |Stirling1(n,k)| * 2^(k-1) * k!.
a(n) ~ n! * exp(n/2) / (4 * (exp(1/2) - 1)^(n+1)). - Vaclav Kotesovec, Jun 29 2019

A384412 Expansion of Product_{k>=1} 1/(1 - k^2 * x)^((1/30) * (2/3)^k).

Original entry on oeis.org

1, 1, 37, 4477, 1139503, 498101431, 332955009307, 315774077663395, 403232260150593946, 667010006578379121074, 1387375789650073950228650, 3544016332332206162590402778, 10907098996548018595779254922854, 39804369748279182675138824291484662, 169958609977149735126105997027662792638
Offset: 0

Views

Author

Seiichi Manyama, May 28 2025

Keywords

Crossrefs

Cf. A004123, A050351, A247082, A384414.

Programs

  • PARI
    b(n) = sum(k=0, n, 2^k*k!*stirling(n, k, 2));
my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, b(2*k)*x^k/k)/10))

Formula

G.f.: exp((1/10) * Sum_{k>=1} b(2*k) * x^k/k), where b(n) = Sum_{k=0..n} 2^k * k! * Stirling2(n,k).

A032111 "BIJ" (reversible, indistinct, labeled) transform of 2,2,2,2...

Original entry on oeis.org

2, 6, 38, 366, 4502, 66606, 1149878, 22687566, 503589782, 12420052206, 336947795318, 9972186170766, 319727684645462, 11039636939221806, 408406422098722358, 16116066766061589966, 675700891505466507542
Offset: 1

Views

Author

Christian G. Bower

Keywords

Links

Crossrefs

Equals A050351(n) + 1. Cf. A004123, A027882.

Formula

E.g.f.: 2(-2+3e^x-e^(2x))/(2-3e^x).

A308440 Matrix product of triangle of Stirling numbers of second kind A008277 and square of unsigned Lah triangle A105278.

Original entry on oeis.org

1, 5, 1, 37, 15, 1, 365, 223, 30, 1, 4501, 3675, 745, 50, 1, 66605, 68071, 18450, 1865, 75, 1, 1149877, 1411515, 479101, 64750, 3920, 105, 1, 22687565, 32512663, 13260030, 2244501, 181650, 7322, 140, 1, 503589781, 825175275, 393017185, 79948050, 8103711, 436590, 12558, 180, 1
Offset: 1

Views

Author

Shuhei Tsujie, May 27 2019

Keywords

Comments

Also the number of k-dimensional flats of the extended Catalan arrangement of dimension n consisting of hyperplanes x_i - x_j = d (1 <= i < j <= n, -2 <= d <= 2).

Examples

			Triangle begins:
     1;
     5,    1;
    37,   15,   1;
   365,  223,  30,  1;
  4501, 3675, 745, 50, 1;
  ...

Links

Crossrefs

Cf. A008277, A105278, A050351 (first column), A109092 (row sums).

Formula

E.g.f.: exp((exp(x)-1)*y/(3-2exp(x))).

A331345 a(n) = (1/n^2) * Sum_{k>=1} k^n * (1 - 1/n)^(k - 1).

Original entry on oeis.org

1, 3, 37, 1015, 48601, 3583811, 376372333, 53343571695, 9808511445361, 2270198126932219, 645790373135121061, 221449391959470686375, 90084675298978081317961, 42890688646618728144279987, 23627228721958495690763944861, 14910259060767841554203065990111
Offset: 1

Views

Author

Ilya Gutkovskiy, Jun 08 2020

Keywords

Links

Crossrefs

Cf. A000670, A050351, A050352, A050353, A094420, A257565, A321189, A334240.

Programs

  • Mathematica
    Join[{1}, Table[1/n^2 Sum[k^n (1 - 1/n)^(k - 1), {k, 1, Infinity}], {n, 2, 16}]]
Table[n! SeriesCoefficient[(Exp[x] - 1)/(Exp[x] - n (Exp[x] - 1)), {x, 0, n}], {n, 1, 16}]

Formula

a(n) = n! * [x^n] (exp(x) - 1) / (exp(x) - n * (exp(x) - 1)).
a(n) = Sum_{k=1..n} Stirling2(n,k) * (n - 1)^(k - 1) * k!.
a(n) ~ sqrt(2*Pi) * n^(2*n - 1/2) / exp(n + 1/2). - Vaclav Kotesovec, Jun 08 2020

A332255 E.g.f.: 1 / (2 - 1 / (2 + x - exp(x))).

Original entry on oeis.org

1, 0, 1, 1, 13, 41, 461, 2745, 32397, 288937, 3794605, 44758649, 665371565, 9660560937, 162652002189, 2782536864697, 52737562595917, 1033546861769513, 21867683869860845, 481630083492884601, 11277805333488014445, 275314710164399079337, 7077059249870048306125
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 08 2020

Keywords

Crossrefs

Cf. A000670, A032032, A050351, A097236.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[1/(2 - 1/(2 + x - Exp[x])), {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    seq(n)={Vec(serlaplace(1/(2 - 1 / (2 + x - exp(x + O(x*x^n))))))} \\ Andrew Howroyd, Feb 08 2020

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * A032032(k) * a(n-k).
a(n) ~ n! * 2^(n-1) / ((c-1) * (2*c-3)^(n+1)), where c = -LambertW(-1, -exp(-3/2)) = 2.3576766739458990584... - Vaclav Kotesovec, Feb 08 2020
Previous Showing 21-27 of 27 results.

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