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 1-4 of 4 results.

A058864 Number of labeled chordal graphs (connected or not) on n nodes with no induced path P_4.

Original entry on oeis.org

1, 2, 8, 49, 402, 4144, 51515, 750348, 12537204, 236424087, 4967735896, 115102258660, 2915655255385, 80164472149454, 2377679022913612, 75674858155603353, 2572626389524849478, 93040490884813025684, 3566833833735159397963, 144485408698878208399296
Offset: 1

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Author

Robert Castelo, Jan 06 2001

Keywords

Comments

A subclass of chordal-comparability graphs.

References

  • Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 417.

Links

Crossrefs

Cf. A007134, A058863, A058865.
Cf. variants: A196555, A196556, A196557.

Programs

  • Mathematica
    Rest[With[{nmax = 50}, CoefficientList[Series[Exp[-LambertW[Exp[-x] - 1]], {x, 0, nmax}], x]*Range[0, nmax]!]] (* G. C. Greubel, Nov 14 2017 *)
a[n_] := Sum[(-1)^(n-k)*StirlingS2[n, k]*(k+1)^(k-1), {k, 0, n}];
Array[a, 18] (* Jean-François Alcover, Dec 17 2017, after Vladeta Jovovic *)
  • PARI
    {a(n)=polcoeff(sum(m=1, n, (m+1)^(m-1)*x^m/prod(k=1, m, 1+k*x+x*O(x^n))), n)} /* Paul D. Hanna, Jul 20 2011 */
  • PARI
    for(n=1,10, print1(sum(k=0, n, (-1)^(n-k)*stirling(n,k,2)*(k+1)^(k-1)), ", ")) \\ G. C. Greubel, Nov 14 2017

Formula

A058863 and A058864 satisfy:
1) c(n) = 1 + Sum_{k=1..n-2} binomial(n, k)*(t(n-k) - c(n-k))
2) t(n) = c(n) + Sum_{k=1..n-1} k*c(k)*binomial(n, k)*t(n-k)/n
where c(n) (A058863) is the number of connected graphs of this type and t(n) (A058864) is the total number of such graphs.
O.g.f.: Sum_{n>=1} (n+1)^(n-1) * x^n / Product_{k=1..n} (1+k*x). - Paul D. Hanna, Jul 20 2011
E.g.f.: exp(-LambertW(exp(-x)-1)). - Vladeta Jovovic, Nov 22 2002
a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling2(n, k)*(k+1)^(k-1). - Vladeta Jovovic, Nov 12 2003
a(n) ~ sqrt(exp(1)-1) * exp(1-n) * n^(n-1) * (1-log(exp(1)-1))^(1/2-n). - Vaclav Kotesovec, Oct 18 2013

Extensions

Formulae edited and completed by Michel Marcus, Apr 07 2013

A196556 O.g.f.: Sum_{n>=0} 3*(n+3)^(n-1) * x^n / Product_{k=1..n} (1+k*x).

Original entry on oeis.org

1, 3, 12, 66, 483, 4476, 50454, 671649, 10328118, 180341094, 3527385345, 76435691250, 1818255212490, 47118807865863, 1321527658352016, 39889359465259446, 1289471521115731611, 44450463108654209136, 1627806562174453037802
Offset: 0

Views

Author

Paul D. Hanna, Oct 03 2011

Keywords

Examples

			O.g.f.: A(x) = 1 + 3*x + 12*x^2 + 66*x^3 + 483*x^4 + 4476*x^5 +...
where the o.g.f. is given by:
A(x) = 1 + 3*4^0*x/(1+x) + 3*5^1*x^2/((1+x)*(1+2*x)) + 3*6^2*x^3/((1+x)*(1+2*x)*(1+3*x)) + 3*7^3*x^4/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)) +...
E.g.f.: A(x) = 1 + 3*x + 12*x^2/2! + 66*x^3/3! + 483*x^4/4! + 4476*x^5/5! +...
where the e.g.f. is given by:
A(x)^(1/3) = 1 + x + 2*x^2/2! + 8*x^3/3! + 49*x^4/4! + 402*x^5/5! + 4144*x^6/6! +...+ A058864(n)*x^n/n! +...

Links

Crossrefs

Cf. A058864, A196555, A196557.

Programs

  • Mathematica
    CoefficientList[Series[E^(-3*LambertW[E^(-x)-1]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jul 09 2013 *)
  • PARI
    {a(n)=polcoeff(sum(m=0, n, 3*(m+3)^(m-1)*x^m/prod(k=1, m, 1+k*x+x*O(x^n))), n)}
  • PARI
    /* E.g.f. = G(x)^3 where G(x) = e.g.f. of A058864 */
{A058864(n)=polcoeff(sum(m=0, n, (m+1)^(m-1)*x^m/prod(k=1, m, 1+k*x+x*O(x^n))), n)}
{a(n)=n!*polcoeff(sum(k=0,n,A058864(k)*x^k/k!+x*O(x^n))^3,n)}
  • PARI
    a(n)=sum(k=0, n, (-1)^(n-k)*stirling(n, k, 2)*3*(k+3)^(k-1));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-3*lambertw(exp(-x)-1)))) \\ Seiichi Manyama, Nov 21 2021

Formula

E.g.f.: exp(-3*LambertW(exp(-x)-1)).
a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling2(n, k)*3*(k+3)^(k-1).
E.g.f.: A(x) = G(x)^3 where G(x) = e.g.f. of A058864, which is the number of labeled chordal graphs (connected or not) on n nodes with no induced path P_4.
a(n) ~ 3*sqrt(exp(1)-1) * n^(n-1) / (exp(n-3) * (1-log(exp(1)-1))^(n-1/2)). - Vaclav Kotesovec, Jul 09 2013

A196557 O.g.f.: Sum_{n>=0} 4*(n+4)^(n-1) * x^n / Product_{k=1..n} (1+k*x).

Original entry on oeis.org

1, 4, 20, 128, 1036, 10308, 122560, 1701092, 27053556, 485683128, 9723771156, 214934627476, 5201286731560, 136818097071820, 3888121468512308, 118737900886653664, 3878569457507036988, 134960059001226137588, 4984357865462772982112
Offset: 0

Views

Author

Paul D. Hanna, Oct 03 2011

Keywords

Examples

			O.g.f.: A(x) = 1 + 4*x + 20*x^2 + 128*x^3 + 1036*x^4 + 10308*x^5 +...
where the o.g.f. is given by:
A(x) = 1 + 4*5^0*x/(1+x) + 4*6^1*x^2/((1+x)*(1+2*x)) + 4*7^2*x^3/((1+x)*(1+2*x)*(1+3*x)) + 4*8^3*x^4/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)) +...
E.g.f.: A(x) = 1 + 4*x + 20*x^2/2! + 128*x^3/3! + 1036*x^4/4! + 10308*x^5/5! +...
where the e.g.f. is given by:
A(x)^(1/2) = 1 + 2*x + 6*x^2/2! + 28*x^3/3! + 186*x^4/4! + 1614*x^5/5! + 17332*x^6/6! +...+ A196555(n)*x^n/n! +...
A(x)^(1/4) = 1 + x + 2*x^2/2! + 8*x^3/3! + 49*x^4/4! + 402*x^5/5! + 4144*x^6/6! +...+ A058864(n)*x^n/n! +...

Links

Crossrefs

Cf. A058864, A196555, A196556.

Programs

  • Mathematica
    CoefficientList[Series[E^(-4*LambertW[E^(-x)-1]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jul 09 2013 *)
  • PARI
    {a(n)=polcoeff(sum(m=0, n, 4*(m+4)^(m-1)*x^m/prod(k=1, m, 1+k*x+x*O(x^n))), n)}
  • PARI
    /* E.g.f. = G(x)^4 where G(x) = e.g.f. of A058864 */
{A058864(n)=polcoeff(sum(m=0, n, (m+1)^(m-1)*x^m/prod(k=1, m, 1+k*x+x*O(x^n))), n)}
{a(n)=n!*polcoeff(sum(k=0,n,A058864(k)*x^k/k!+x*O(x^n))^4,n)}
  • PARI
    a(n)=sum(k=0, n, (-1)^(n-k)*stirling(n, k, 2)*4*(k+4)^(k-1));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-4*lambertw(exp(-x)-1)))) \\ Seiichi Manyama, Nov 21 2021

Formula

E.g.f.: exp(-4*LambertW(exp(-x)-1)).
a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling2(n, k)*4*(k+4)^(k-1).
E.g.f.: A(x) = G(x)^4 where G(x) = e.g.f. of A058864, which is the number of labeled chordal graphs (connected or not) on n nodes with no induced path P_4.
a(n) ~ 4*sqrt(exp(1)-1)*n^(n-1)/(exp(n-4)*(1-log(exp(1)-1))^(n-1/2)). - Vaclav Kotesovec, Jul 09 2013

A349527 a(n) = Sum_{k=0..n} (-1)^(n-k) * (2*k+1)^(k-1) * Stirling2(n,k).

Original entry on oeis.org

1, 1, 4, 35, 469, 8502, 194807, 5402497, 175985390, 6587650497, 278674144201, 13148017697608, 684554867667117, 38988819551585477, 2411411875573335044, 160951864352781351959, 11531509389384310870257
Offset: 0

Views

Author

Seiichi Manyama, Nov 20 2021

Keywords

Links

Crossrefs

Cf. A008277, A058864, A196555, A349524, A349528.

Programs

  • Mathematica
    a[n_] := Sum[(-1)^(n - k) * (2*k + 1)^(k - 1) * StirlingS2[n, k], {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, Nov 27 2021 *)
  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k)*(2*k+1)^(k-1)*stirling(n, k, 2));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(2*(exp(-x)-1))/2)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (2*k+1)^(k-1)*x^k/prod(j=1, k, 1+j*x)))

Formula

E.g.f. satisfies: log(A(x)) = (1 - exp(-x)) * A(x)^2.
E.g.f.: exp( -LambertW(2 * (exp(-x) - 1))/2 ).
G.f.: Sum_{k>=0} (2*k+1)^(k-1) * x^k/Product_{j=1..k} (1 + j*x).
a(n) ~ sqrt(2*exp(1) - 1) * sqrt(log(2/(2-exp(-1)))) * n^(n-1) / (2 * exp(n - 1/2) * (1 + log(2/(2*exp(1) - 1)))^n). - Vaclav Kotesovec, Nov 21 2021
Showing 1-4 of 4 results.

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