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.

A335344 Expansion of e.g.f. exp(x^2/(2*(1 - x)^2)).

Original entry on oeis.org

1, 0, 1, 6, 39, 300, 2715, 28350, 335265, 4422600, 64298745, 1020477150, 17542820295, 324552648420, 6426708843555, 135573281994150, 3034400481137025, 71801374285040400, 1790523094644709425, 46923435009924823350, 1289032229351717425575
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 02 2020

Keywords

Comments

For n>0, a(n) is also the number of ways to split n people into nonempty groups, have each group sit around a circular table, and select 2 people from each table (where two seating arrangements are considered identical if each person has the same left neighbors in both of them). See example below. - Enrique Navarrete, Oct 01 2023

Examples

			For n = 5, using one table, there are 4! circular seatings and binomial(5,2) ways to select 2 persons, hence 240 ways. Using two tables, the only way we can select 2 persons from each one is seating 3 persons in one table and 2 in the other, which can be done in 20 ways; then choosing 2 persons from each table can be done in 3 ways, hence giving another 60 ways for a total of 300. - _Enrique Navarrete_, Oct 01 2023

Links

Crossrefs

Cf. A001286, A052887, A082579, A185369, A335345.

Programs

  • Maple
    f:= gfun:-rectoproc({a(n) = -(n-1)*(3*n-7)*a(n-2) + 3*(n-1)*a(n-1) + (n - 1)*(n - 2)*(n - 3)*a(n-3),a(0)=1,a(1)=0,a(2)=1},a(n),remember):
map(f, [$0..30]); # Robert Israel, Jun 04 2020
  • Mathematica
    nmax = 20; CoefficientList[Series[Exp[x^2/(2 (1 - x)^2)], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = (1/2) Sum[Binomial[n - 1, k - 1] (k - 1) k! a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
  • PARI
    seq(n)=Vec(serlaplace(exp(x^2/(2*(1 - x)^2) + O(x*x^n)))) \\ Andrew Howroyd, Jun 02 2020

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A001286(k) * a(n-k).
D-finite with recurrence a(n) = -(n - 1)*(3*n - 7)*a(n - 2) + 3*(n - 1)*a(n - 1) + (n - 1)*(n - 2)*(n - 3)*a(n - 3). - Robert Israel, Jun 04 2020
a(n) ~ n^(n - 1/6) * exp(1/6 - n^(1/3)/2 + 3*n^(2/3)/2 - n) / sqrt(3). - Vaclav Kotesovec, Jun 11 2020
a(n) = n! * Sum_{k=0..floor(n/2)} binomial(n-1,n-2*k)/(2^k * k!). - Seiichi Manyama, Mar 16 2023

A361533 Expansion of e.g.f. exp(x^3/(6 * (1-x))).

Original entry on oeis.org

1, 0, 0, 1, 4, 20, 130, 980, 8400, 80920, 865200, 10164000, 130114600, 1802600800, 26867640800, 428661633400, 7288513232000, 131558835408000, 2512282795422400, 50600743739145600, 1071998968264224000, 23829055696093648000, 554524256514356128000
Offset: 0

Views

Author

Seiichi Manyama, Mar 15 2023

Keywords

Crossrefs

Cf. A185369, A361545, A361547.
Cf. A293049.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^3/(6*(1-x)))))

Formula

a(n) = 2*(n-1) * a(n-1) - (n-1)*(n-2) * a(n-2) + binomial(n-1,2) * a(n-3) - 2*binomial(n-1,3) * a(n-4) for n > 3.
a(n) ~ 2^(-3/4) * 3^(-1/4) * exp(-5/12 + sqrt(2*n/3) - n) * n^(n - 1/4). - Vaclav Kotesovec, Mar 29 2023
From Seiichi Manyama, Jun 17 2024: (Start)
a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n-2*k-1,n-3*k)/(6^k * k!).
a(0) = 1; a(n) = ((n-1)!/6) * Sum_{k=3..n} k * a(n-k)/(n-k)!. (End)

A361545 Expansion of e.g.f. exp(x^4/(24 * (1-x))).

Original entry on oeis.org

1, 0, 0, 0, 1, 5, 30, 210, 1715, 15750, 160650, 1801800, 22043175, 292116825, 4168464300, 63725161500, 1039028615625, 17998106626500, 330068683444500, 6388785205803000, 130156170633113625, 2783924007745505625, 62375052003905891250, 1460924768552182683750
Offset: 0

Views

Author

Seiichi Manyama, Mar 15 2023

Keywords

Crossrefs

Cf. A185369, A361533, A361547.
Cf. A293050.

Programs

  • Mathematica
    RecurrenceTable[{3 (-4 + n) (-3 + n) (-2 + n) (-1 + n) a[-5 + n] - 4 (-3 + n) (-2 + n) (-1 + n) a[-4 + n] + 24 (-2 + n) (-1 + n) a[-2 + n] - 48 (-1 + n) a[-1 + n] + 24 a[n] == 0, a[1] == 0, a[2] == 0, a[3] == 0, a[4] == 1, a[5] == 5}, a, {n, 0, 25}] (* Vaclav Kotesovec, Aug 28 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^4/(24*(1-x)))))

Formula

a(n) = 2*(n-1) * a(n-1) - (n-1)*(n-2) * a(n-2) + binomial(n-1,3) * a(n-4) - 3*binomial(n-1,4) * a(n-5) for n > 4.
From Seiichi Manyama, Jun 17 2024: (Start)
a(n) = n! * Sum_{k=0..floor(n/4)} binomial(n-3*k-1,n-4*k)/(24^k * k!).
a(0) = 1; a(n) = ((n-1)!/24) * Sum_{k=4..n} k * a(n-k)/(n-k)!. (End)
a(n) ~ 2^(-5/4) * 3^(-1/4) * exp(-7/48 + sqrt(n/6) - n) * n^(n - 1/4). - Vaclav Kotesovec, Aug 28 2025

A335345 Expansion of e.g.f. exp(x^2/(2*(1 - x)^3)).

Original entry on oeis.org

1, 0, 1, 9, 75, 690, 7305, 89145, 1237425, 19221300, 329371245, 6157738125, 124551652995, 2707913238030, 62945320162725, 1557291398788125, 40844991621859425, 1131753403094113800, 33025920511859300025, 1012128709342410284625, 32494107983067177522075
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 02 2020

Keywords

Crossrefs

Cf. A001809, A091695, A185369, A335344.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Exp[x^2/(2 (1 - x)^3)], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = (1/4) Sum[Binomial[n - 1, k - 1] k (k - 1) k! a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
  • PARI
    seq(n)=Vec(serlaplace(exp(x^2/(2*(1 - x)^3) + O(x*x^n)))) \\ Andrew Howroyd, Jun 02 2020

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A001809(k) * a(n-k).
D-finite with recurrence 2*a(n) +8*(-n+1)*a(n-1) +2*(n-1)*(6*n-13)*a(n-2) -(n-1)*(n-2)*(8*n-23)*a(n-3) +2*(n-1)*(n-2)*(n-3)*(n-4)*a(n-4)=0. - R. J. Mathar, Jun 05 2020
a(n) ~ 2^(-9/8) * 3^(1/8) * n^(n - 1/8) * exp(1/54 - n^(1/4)/(2^(15/4)*3^(5/4)) - sqrt(6*n)/12 + 2^(7/4)*3^(-3/4)*n^(3/4) - n). - Vaclav Kotesovec, Jun 11 2020
a(n) = n! * Sum_{k=0..floor(n/2)} binomial(n+k-1,n-2*k)/(2^k * k!). - Seiichi Manyama, Jun 17 2024

A361532 Expansion of e.g.f. exp((x + x^2/2)/(1-x)).

Original entry on oeis.org

1, 1, 4, 19, 118, 886, 7786, 78184, 881644, 11017108, 150966856, 2249261356, 36181351504, 624658612384, 11516406883528, 225740649754936, 4686671645814736, 102712289940757264, 2369128149877075264, 57359541280704038128, 1454229915957292684576
Offset: 0

Views

Author

Seiichi Manyama, Mar 15 2023

Keywords

Crossrefs

Cf. A361548, A361557, A361558.
Cf. A052845, A112242, A185369.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[Exp[(x+x^2/2)/(1-x)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 08 2023 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((x+x^2/2)/(1-x))))

Formula

a(n) = (2*n-1) * a(n-1) - (n-1)*(n-3) * a(n-2) - binomial(n-1,2) * a(n-3) for n > 2.
a(n) ~ 2^(-3/4) * 3^(1/4) * exp(-5/4 + sqrt(6*n) - n) * n^(n - 1/4) * (1 + sqrt(3)/(2*sqrt(2*n))). - Vaclav Kotesovec, Mar 20 2023

A361547 Expansion of e.g.f. exp(x^5/(120 * (1-x))).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 6, 42, 336, 3024, 30366, 335412, 4041576, 52756704, 741620880, 11169844686, 179448036768, 3063069801792, 55360031126400, 1056123043335360, 21208345049147256, 447183762148547424, 9877939209960101280, 228112734232663600320
Offset: 0

Views

Author

Seiichi Manyama, Mar 15 2023

Keywords

Comments

In general, if m>=1 and e.g.f. = exp(x^m / (m! * (1-x))), then a(n) ~ n! * exp(2*sqrt(n/m!) - (2*m-1)/(2*m!)) / (2*sqrt(Pi) * m!^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 28 2025

Crossrefs

Cf. A185369, A361533, A361545.

Programs

  • Mathematica
    RecurrenceTable[{4 (-5 + n) (-4 + n) (-3 + n) (-2 + n) (-1 + n) a[-6 + n] - 5 (-4 + n) (-3 + n) (-2 + n) (-1 + n) a[-5 + n] + 120 (-2 + n) (-1 + n) a[-2 + n] - 240 (-1 + n) a[-1 + n] + 120 a[n] == 0, a[1] == 0, a[2] == 0, a[3] == 0, a[4] == 0, a[5] == 1, a[6] == 6}, a, {n, 0, 25}] (* Vaclav Kotesovec, Aug 28 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^5/(120*(1-x)))))

Formula

a(n) = 2*(n-1) * a(n-1) - (n-1)*(n-2) * a(n-2) + binomial(n-1,4) * a(n-5) - 4*binomial(n-1,5) * a(n-6) for n > 5.
From Seiichi Manyama, Jun 17 2024: (Start)
a(n) = n! * Sum_{k=0..floor(n/5)} binomial(n-4*k-1,n-5*k)/(120^k * k!).
a(0) = 1; a(n) = ((n-1)!/120) * Sum_{k=5..n} k * a(n-k)/(n-k)!. (End)
a(n) ~ 2^(-5/4) * 15^(-1/4) * exp(-3/80 + sqrt(n/30) - n) * n^(n - 1/4). - Vaclav Kotesovec, Aug 28 2025

A330041 Expansion of e.g.f. exp(cosh(exp(x) - 1) - 1).

Original entry on oeis.org

1, 0, 1, 3, 11, 55, 322, 2114, 15556, 127005, 1135374, 11011220, 115080825, 1288589757, 15379512670, 194796087841, 2608470709562, 36805935282625, 545626818921885, 8475730766054047, 137637670315066835, 2331584745107027528, 41122505417366272200
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 28 2019

Keywords

Comments

Stirling transform of A005046 (with interpolated zeros).
Exponential transform of A024430.

Links

Crossrefs

Cf. A000258, A005046, A024430, A185369, A330021.

Programs

  • Maple
    g:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(2*n-1, 2*k-1) *g(n-k), k=1..n))
    end:
b:= proc(n, m) option remember; `if`(n=0,
     `if`(m::odd, 0, g(m/2)), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..22);  # Alois P. Heinz, Jun 23 2023
  • Mathematica
    nmax = 22; CoefficientList[Series[Exp[Cosh[Exp[x] - 1] - 1], {x, 0, nmax}], x] Range[0, nmax]!

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A024430(k) * a(n-k).

A373770 Expansion of e.g.f. exp(x^2 / (2 * (1 - x))) / (1 - x).

Original entry on oeis.org

1, 1, 3, 12, 63, 405, 3075, 26880, 265545, 2922885, 35447895, 469396620, 6736095135, 104102463465, 1723322736135, 30416726340000, 570089983287825, 11306156398562025, 236514323713142475, 5204122351983254700, 120139520273298100575, 2903216115946088267325
Offset: 0

Views

Author

Seiichi Manyama, Jun 18 2024

Keywords

Crossrefs

Cf. A130905, A361596, A373771.
Cf. A185369.

Programs

  • PARI
    a(n) = n!*sum(k=0, n\2, binomial(n-k, n-2*k)/(2^k*k!));

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} binomial(n-k,n-2*k)/(2^k * k!).
From Vaclav Kotesovec, Jun 18 2024: (Start)
Recurrence: 2*a(n) = 2*(2*n-1)*a(n-1) - 2*(n-2)*(n-1)*a(n-2) - (n-2)*(n-1)*a(n-3).
a(n) ~ 2^(-1/4) * exp(-3/4 + sqrt(2*n) - n) * n^(n + 1/4) * (1 + 7/(6*sqrt(2*n))). (End)
Showing 1-8 of 8 results.

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