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.

A003514 Number of series-reduced labeled graphs with n nodes.

Original entry on oeis.org

1, 1, 2, 4, 15, 102, 4166, 402631, 76374899, 27231987762, 18177070202320, 22801993267433275, 54212469444212172845, 246812697326518127351384, 2173787304796735262709419350, 37373588848096468764431235680525, 1263513534110606141026676778422031561
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Row sums of A060514 and A307806.
The unlabeled version is A005637.
Cf. A003515 (connected).

Programs

  • Mathematica
    max = 15; f[x_] := (1 + x)^(-1/2)*Exp[x/2-x^2/4]*Sum[(2*Exp[-x/(1+x)])^Binomial[k, 2]*Exp[x^2/2/(1+x)]^k*x^k/k!, {k, 0, max}]; CoefficientList[ Series[f[x], {x, 0, max}], x]*Range[0, max]!(* Jean-François Alcover, Nov 25 2011, after Vladeta Jovovic *)
  • PARI
    seq(n)={my(x='x+O('x^(n+1))); Vec(serlaplace((1 + x)^( - 1/2) * exp(x/2 - x^2/4) * sum(k=0, n, (2 * exp(-x/(1 + x)))^binomial(k, 2) * (exp(x^2/2/(1 + x)))^k * x^k/k!)))} \\ Andrew Howroyd, Feb 23 2024

Formula

E.g.f.: (1 + x)^( - 1/2) * exp(x/2 - x^2/4) * Sum_{k=0..inf} (2 * exp( - x/(1 + x)))^binomial(k, 2) * (exp(x^2/2/(1 + x)))^k * x^k/k!. - Vladeta Jovovic, Mar 23 2001

Extensions

More terms from Vladeta Jovovic, Mar 23 2001

A331437 Triangle read by rows: T(n,k) = number of homeomorphically irreducible connected labeled graphs with n edges and k vertices, n >= 0, 1 <= k <= n+1.

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 96, 0, 0, 0, 1, 0, 120, 427, 0, 0, 0, 0, 20, 180, 1260, 6448, 0, 0, 0, 0, 15, 420, 3780, 23520, 56961, 0, 0, 0, 0, 10, 700, 10850, 79800, 347760, 892720, 0, 0, 0, 0, 1, 837, 24045, 269360, 1655640, 6400800, 11905091
Offset: 0

Views

Author

N. J. A. Sloane, Jan 19 2020

Keywords

Comments

Homeomorphically irreducible graphs are graphs without vertices of degree 2. - Andrew Howroyd, Jan 24 2020

Examples

			Triangle begins:
  1;
  0, 1;
  0, 0, 0;
  0, 0, 0, 4;
  0, 0, 0, 0,  5;
  0, 0, 0, 0,  0,  96;
  0, 0, 0, 1,  0, 120,  427;
  0, 0, 0, 0, 20, 180, 1260,  6448;
  0, 0, 0, 0, 15, 420, 3780, 23520, 56961;
...

Links

Crossrefs

Column sums are A003515.
Row sums are A331584.
Right diagonal is A005512(n+1).
Cf. A060514, A331438 (transpose).

Programs

  • PARI
    \\ See Jackson & Reilly for e.g.f.
H(n,y) = {my(A=O(x*x^n)); (exp(y*x/2 - (y*x)^2/4 + A)/sqrt(1 + y*x + A))*sum(k=0, n, ((1 + y)*exp(-y^2*x/(1+y*x) + A))^binomial(k,2) * (x*exp((y^3*x^2 + A)/(2*(1 + y*x))))^k / k!)}
T(n) = {Mat([Col(p, -n) | p<-Vec(serlaplace(log(H(n,y + O(y^n)))))])}
{ my(A=T(10)); for(n=1, #A, print(A[n, 1..n])) } \\ Andrew Howroyd, Jan 24 2020

Extensions

Terms a(44) and beyond from Andrew Howroyd, Jan 24 2020

A331438 Irregular triangle read by rows: T(n,k) = number of homeomorphically irreducible connected labeled graphs with n vertices and k edges, n >= 1, 0 <= k <= n*(n-1)/2.

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 1, 0, 0, 0, 0, 5, 0, 0, 20, 15, 10, 1, 0, 0, 0, 0, 0, 96, 120, 180, 420, 700, 837, 765, 395, 105, 15, 1, 0, 0, 0, 0, 0, 0, 427, 1260, 3780, 10850, 24045, 44814, 68040, 80955, 70500, 43232, 18774, 5880, 1330, 210, 21, 1, 0, 0, 0, 0, 0, 0, 0, 6448, 23520, 79800, 269360, 782880, 1956136, 4203360, 7610340, 11365676
Offset: 1

Views

Author

N. J. A. Sloane, Jan 19 2020

Keywords

Examples

			Triangle begins:
1,
0,1,
0,0,0,0,
0,0,0,4,0,0,1,
0,0,0,0,5,0,0,20,15,10,1,
0,0,0,0,0,96,120,180,420,700,837,765,395,105,15,1,
0,0,0,0,0,0,427,1260,3780,10850,24045,44814,68040,80955,70500,43232,18774,5880,1330,210,21,1,
0,0,0,0,0,0,0,6448,23520,79800,269360,782880,1956136,4203360,7610340,11365676,...,
...

Links

Crossrefs

Row sums are A003515.
Cf. A060514, A331437 (transpose).

Programs

  • PARI
    \\ See Jackson & Reilly for e.g.f.
H(n,y)={my(A=O(x*x^n)); (exp(y*x/2 - (y*x)^2/4 + A)/sqrt(1 + y*x + A))*sum(k=0, n, ((1 + y)*exp(-y^2*x/(1+y*x) + A))^binomial(k,2) * (x*exp((y^3*x^2 + A)/(2*(1 + y*x))))^k / k!)}
Row(n)={Vecrev(n!*polcoef(log(H(n,y)), n), binomial(n,2)+1)}
{ for(n=1, 6, print(Row(n))) } \\ Andrew Howroyd, Jan 24 2020

A060517 Triangle T(n,k) of series-reduced (or homeomorphically irreducible) graphs with loops on n labeled nodes and with k edges, k=0..binomial(n+1,2).

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 1, 1, 3, 6, 6, 6, 3, 1, 1, 6, 15, 34, 58, 60, 60, 50, 33, 10, 1, 1, 10, 35, 120, 265, 475, 820, 1200, 1615, 1860, 1693, 1060, 425, 105, 15, 1, 1, 15, 75, 330, 990, 2691, 6326, 13170, 26205, 48055, 79206, 112863, 133535, 124680, 88890, 47874
Offset: 0

Views

Author

Vladeta Jovovic, Mar 24 2001

Keywords

Examples

			[1], [1, 0], [1, 1, 2, 1], [1, 3, 6, 6, 6, 3, 1], [1, 6, 15, 34, 58, 60, 60, 50, 33, 10, 1], [1, 10, 35, 120, 265, 475, 820, 1200, 1615, 1860, 1693, 1060, 425, 105, 15, 1], [1, 15, 75, 330, 990, 2691, 6326, 13170, 26205, 48055, 79206, 112863, 133535, 124680, 88890, 47874, 19443, 5925, 1330, 210, 21, 1], ...

References

  • I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.

Crossrefs

Row sums: A060516, A003514, A060514.

Formula

E.g.f.: (1 + x * y)^( - 1/2) * exp( - x * y/2 - x^2 * y^2/4) * Sum_{k=0..inf}(1 + x)^binomial(k + 1, 2) * exp( - x^2 * y * k^2/(2 * (1 + x * y)) + x^2 * y * k/2) * x^k/k!
Showing 1-4 of 4 results.

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

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