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 1-6 of 6 results.

A185369 Number of simple labeled graphs on n nodes of degree 1 or 2 without cycles.

Original entry on oeis.org

1, 0, 1, 3, 15, 90, 645, 5355, 50505, 532980, 6219045, 79469775, 1103335695, 16533226710, 265888247625, 4566885297975, 83422361847825, 1614626682669000, 33003508539026025, 710350201433547675, 16057073233633006575
Offset: 0

Views

Author

Geoffrey Critzer, Feb 20 2011

Keywords

Examples

			a(4) = 15 because there are 15 simple labeled graphs on 4 nodes of degree 1 or 2 without cycles: 1-2 3-4, 1-3 2-4, 1-4 2-3, 1-2-3-4, 1-2-4-3, 1-3-2-4, 1-3-4-2, 1-4-2-3, 1-4-3-2, 2-1-3-4, 2-1-4-3, 3-1-2-4, 3-1-4-2, 4-1-2-3, 4-1-3-2.

References

  • Herbert S. Wilf, Generatingfunctionology, p. 104.

Links

Crossrefs

Cf. A335344, A335345.
Cf. A361533, A361545, A361547.

Programs

  • Maple
    a:= proc(n) option remember;
       `if`(n<2, 1-n, add(binomial(n-1, k-1) *k!/2 *a(n-k), k=2..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Feb 24 2011
  • Mathematica
    a=1/(2(1-x))-1/2-x/2;
Range[0,20]! CoefficientList[Series[Exp[a],{x,0,20}],x]

Formula

E.g.f.: exp(1/(2*(1-x))-x/2-1/2).
a(n) = 1-n if n<2, else a(n) = Sum_{k=2..n} C(n-1,k-1) * k!/2 * a(n-k).
a(n) ~ 2^(-3/4)*n^(n-1/4)*exp(-3/4+sqrt(2*n)-n). - Vaclav Kotesovec, Sep 25 2013
Conjecture: +2*a(n) +4*(-n+1)*a(n-1) +2*(n-1)*(n-3)*a(n-2) +(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Jun 14 2016
a(n) = n! * Sum_{k=0..floor(n/2)} binomial(n-k-1,n-2*k)/(2^k * k!). - Seiichi Manyama, Jun 17 2024

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)

A293050 Expansion of e.g.f. exp(x^4/(1 - x)).

Original entry on oeis.org

1, 0, 0, 0, 24, 120, 720, 5040, 60480, 725760, 9072000, 119750400, 1756339200, 28021593600, 479480601600, 8717829120000, 168254102016000, 3438311804928000, 74160828758016000, 1682757222322176000, 40061786401308672000, 998402161605488640000
Offset: 0

Views

Author

Seiichi Manyama, Sep 29 2017

Keywords

Links

Crossrefs

Column k=3 of A293053.
E.g.f.: Product_{i>k} exp(x^i): A000262 (k=0), A052845 (k=1), A293049 (k=2), this sequence (k=3).
Cf. A361545, A361576.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n-1, j-1)*j!, j=4..n))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 29 2017
  • Mathematica
    a[n_] := a[n] = If[n==0, 1, Sum[a[n-j] Binomial[n-1, j-1] j!, {j, 4, n}]];
a /@ Range[0, 23] (* Jean-François Alcover, Dec 21 2020, after Alois P. Heinz *)
  • PARI
    x='x+O('x^66); Vec(serlaplace(exp(x^4/(1-x))))

Formula

E.g.f.: Product_{i>3} exp(x^i).
From Vaclav Kotesovec, Sep 30 2017: (Start)
a(n) = 2*(n-1)*a(n-1) - (n-2)*(n-1)*a(n-2) + 4*(n-3)*(n-2)*(n-1)*a(n-4) - 3*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5).
a(n) ~ n^(n-1/4) * exp(-7/2 + 2*sqrt(n) - n) / sqrt(2).
(End)
From Seiichi Manyama, Jun 17 2024: (Start)
a(n) = n! * Sum_{k=0..floor(n/4)} binomial(n-3*k-1,n-4*k)/k!.
a(0) = 1; a(n) = (n-1)! * Sum_{k=4..n} k * a(n-k)/(n-k)!. (End)

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

A373758 Expansion of e.g.f. exp(x^4/(24 * (1 - x)^2)).

Original entry on oeis.org

1, 0, 0, 0, 1, 10, 90, 840, 8435, 91980, 1089900, 13998600, 194184375, 2897744850, 46335539250, 790936146000, 14361717995625, 276491756541000, 5626652076045000, 120696581303298000, 2722068344529158625, 64392333741216731250, 1594243471325576321250
Offset: 0

Views

Author

Seiichi Manyama, Jun 17 2024

Keywords

Crossrefs

Cf. A361545, A361577, A373759.

Programs

  • PARI
    a(n) = n!*sum(k=0, n\4, binomial(n-2*k-1, n-4*k)/(24^k*k!));
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!/24*sum(j=4, i, j*(j-3)*v[i-j+1]/(i-j)!)); v;

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} binomial(n-2*k-1,n-4*k)/(24^k * k!).
a(0) = 1; a(n) = ((n-1)!/24) * Sum_{k=4..n} k * (k-3) * a(n-k)/(n-k)!.

A373759 Expansion of e.g.f. exp(x^4/(24 * (1 - x)^3)).

Original entry on oeis.org

1, 0, 0, 0, 1, 15, 180, 2100, 25235, 319410, 4299750, 61815600, 950524575, 15633092475, 274749725250, 5151569172750, 102831791687625, 2179782464359500, 48933251188321500, 1160002995644493000, 28956069155772383625, 759014081927743516875
Offset: 0

Views

Author

Seiichi Manyama, Jun 17 2024

Keywords

Crossrefs

Cf. A361545, A361577, A373758.

Programs

  • PARI
    a(n) = n!*sum(k=0, n\4, binomial(n-k-1, n-4*k)/(24^k*k!));
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!/24*sum(j=4, i, j*binomial(j-2, j-4)*v[i-j+1]/(i-j)!)); v;

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} binomial(n-k-1,n-4*k)/(24^k * k!).
a(0) = 1; a(n) = ((n-1)!/24) * Sum_{k=4..n} k * binomial(k-2,k-4) * a(n-k)/(n-k)!.
Showing 1-6 of 6 results.

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