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.

A048174 Number of labeled chains with n edges.

Original entry on oeis.org

1, 1, 7, 73, 1051, 19381, 436087, 11585953, 354981571, 12322179901, 477938035807, 20485584143113, 961567521142411, 49054912287659461, 2702571588828034567, 159911968233095867953, 10114120854154243738771, 680943323845807848142861, 48622150270026820216099567, 3670113810844512283440027673
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of labeled series-parallel posets on n nodes that are not a nontrivial ordinal sum.
Let ( T, < ) and ( U, < ) be posets with T and U disjoint. Their ordinal sum is ( T union U, < ) where x

References

  • R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.39, page 133, h(n).

Links

Programs

  • Maple
    with(gfun): f := series(ln(1+x)-x^2/(1+x), x, 30):
egf := seriestoseries(f, 'revogf'):
t := series(egf/(1+egf), x, 21):
seriestolist(t, 'Laplace');
  • Mathematica
    lim = 20; Drop[ CoefficientList[ InverseSeries[ Series[-Log[1 - x] - x^2/(1 - x), {x, 0, lim}], y], y], 1]*Range[lim]! (* Jean-François Alcover, Sep 21 2011, after g.f. *)
max = 18; S053554 = InverseSeries[ Series[ Log[1+x] - x^2/(1+x), {x, 0, max}], x]; Drop[ CoefficientList[ Series[ S053554 / (1+S053554), {x, 0, max}], x]* Range[0, max]!, 1] (* Jean-François Alcover, Nov 29 2011, after Maple *)
  • Maxima
    a(n):=if n=1 then 1 else (sum((n+k-1)!*sum((-1)^(j)/(k-j)!*((sum(sum((binomial(-l+i-1,l-1)*(-1)^(n-i-1)*stirling1(n+j-i-1,j-l))/(l!*(n+j-i-1)!),i,2*l,n+l-1),l,1,j))+((-1)^(n-1)*stirling1(n+j-1,j))/(n+j-1)!),j,1,k),k,1,n-1)); /* Vladimir Kruchinin, Feb 19 2012 */
  • PARI
    x='x+O('x^66); s=serreverse(log(1+x)-x^2/(1+x)); Vec(serlaplace(s/(1+s))) \\ Joerg Arndt, Mar 11 2014

Formula

Reference gives generating functions (see PARI code for one example).
A048172(n) = A058349(n) + a(n), n>1.
A053554(n) = A058349(n) + A058350(n) (n>=2).
a(n)=sum(k=1..n-1, (n+k-1)!*sum(j=1..k, (-1)^(j)/(k-j)!*((sum(l=1..j, sum(i=2*l..n+l-1, (binomial(-l+i-1,l-1)*(-1)^(n-i-1)*stirling1(n+j-i-1,j-l))/(l!*(n+j-i-1)!))))+((-1)^(n-1)*stirling1(n+j-1,j))/(n+j-1)!))). - Vladimir Kruchinin, Feb 19 2012
a(n) ~ (5-sqrt(5)) * n^(n-1) / (2*5^(3/4)*exp(n)*(2-sqrt(5)+log((1+sqrt(5))/2))^(n-1/2)). - Vaclav Kotesovec, Mar 09 2014

Extensions

More terms from Joerg Arndt, Feb 04 2011

A058349 Number of connected labeled series-parallel posets on n nodes.

Original entry on oeis.org

1, 2, 12, 122, 1740, 31922, 715932, 18978122, 580513260, 20125554242, 779832497532, 33398722757402, 1566656717322060, 79879485803841362, 4398701789915269212, 260166428897541369962, 16449181879032096013740, 1107112451498156565581282, 79030557433744270179981372
Offset: 1

Views

Author

N. J. A. Sloane, Dec 16 2000

Keywords

Comments

Also, number of labeled blobs with n edges.

References

  • R. C. Read, Graphical enumeration by cycle-index sums: first steps toward a unified treatment, preprint, Sept. 26, 1991.
  • R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.39, page 133, g(n).

Links

Crossrefs

A053554(n) = a(n) + A058350(n) (n>=2).

Programs

  • Maple
    (continue from A053554) t1 := log(1+EGF053554): t2 := series(t1,x,30); SERIESTOLISTMULT(t2);
  • Mathematica
    Drop[ CoefficientList[ InverseSeries[ Series[x + 2*(1 - Cosh[x]) , {x, 0, 19}], y], y], 1]* Range[19]! (* Jean-François Alcover, Sep 21 2011, after g.f. *)
  • Maxima
    a(n):=if n=1 then 1 else (n-1)!*sum(binomial(n+k-1,n-1)*sum(binomial(k,j)*((sum((binomial(j,l)*((-1)^(n-l+j-1)+1)*sum(binomial(j-l,r)*2^(j-l-r-1)*(-1)^(r-j)*sum((r-2*i)^(n-l+j-1)*binomial(r,i),i,0,r),r,1,j-l))/(n-l+j-1)!,l,0,j-1))),j,1,k),k,1,n-1); /* Vladimir Kruchinin, Feb 19 2012 */
  • PARI
    /* Joerg Arndt, Feb 04 2011 */
x='x+O('x^55); t=x+2*(1-cosh(x));
Vec(serlaplace(serreverse(t))) /* show terms */

Formula

Read (1991) reference gives generating functions (see PARI code for one example).
A048172(n) = a(n)+A048174(n), n>1.
a(n) = (n-1)!*sum(k=1..n-1, binomial(n+k-1,n-1)*sum(j=1..k, binomial(k,j)*((sum(l=0..j-1, (binomial(j,l)*((-1)^(n-l+j-1)+1)*sum(r=1..j-l, binomial(j-l,r)*2^(j-l-r-1)*(-1)^(r-j)*sum(i=0..r, (r-2*i)^(n-l+j-1)*binomial(r,i))))/(n-l+j-1)!))))), n>1, a(1)=1. - Vladimir Kruchinin, Feb 19 2012
a(n) ~ n^(n-1) / (5^(1/4)*exp(n)*(2-sqrt(5)+log((1+sqrt(5))/2))^(n-1/2)). - Vaclav Kotesovec, Mar 09 2014

Extensions

More terms from Joerg Arndt, Feb 04 2011

A202180 Number of n-element unlabeled connected N-free posets.

Original entry on oeis.org

1, 1, 3, 9, 31, 115, 474, 2097, 9967, 50315, 268442, 1505463, 8840306, 54169431
Offset: 1

Views

Author

N. J. A. Sloane, Dec 13 2011

Keywords

Links

Crossrefs

Row sums of A202178 and A202179.
Cf. A003430, A007453, A053554.

Extensions

Missing term a(12) inserted by Salah Uddin Mohammad, May 26 2020

A202181 Triangle read by rows: T(n,k) = number of n-element unlabeled N-free posets of height k (1 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 6, 1, 1, 13, 24, 10, 1, 1, 25, 77, 61, 15, 1, 1, 43, 228, 291, 130, 21, 1, 1, 76, 644, 1229, 856, 246, 28, 1, 1, 128, 1776, 4872, 4840, 2136, 427, 36, 1, 1, 216, 4854, 18711, 25107, 15543, 4733, 694, 45, 1, 1, 354, 13184, 70858, 124167, 101538, 43120, 9577, 1071, 55, 1
Offset: 1

Views

Author

N. J. A. Sloane, Dec 13 2011

Keywords

Examples

			Triangle begins:
1
1 1
1 3 1
1 7 6 1
1 13 24 10 1
1 25 77 61 15 1
1 43 228 291 130 21 1
1 76 644 1229 856 246 28 1
1 128 1776 4872 4840 2136 427 36 1
1 216 4854 18711 25107 15543 4733 694 45 1
1 354 13184 70858 124167 101538 43120 9577 1071 55 1
...

Links

Crossrefs

Row sums give A202182. Cf. A202178, A003430, A007453, A053554.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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