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.

Previous Showing 11-17 of 17 results.

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!

A060535 Number of homeomorphically irreducible multigraphs (or series-reduced multigraphs or multigraphs without nodes of degree 2) on 5 labeled nodes.

Original entry on oeis.org

1, 10, 15, 30, 165, 430, 1170, 3180, 7935, 18610, 40948, 84570, 164740, 304690, 538630, 915574, 1504135, 2398460, 3725495, 5653790, 8404075, 12261860, 17592335, 24857870, 34638440, 47655326, 64798470, 87157890, 116059590
Offset: 0

Views

Author

Vladeta Jovovic, Apr 01 2001

Keywords

References

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

Links

Crossrefs

Cf. A003514, A060516, A060533-A060537.

Formula

G.f.: (5*x^18 - 20*x^17 + 30*x^16 + 58*x^15 - 745*x^14 + 2790*x^13 - 5270*x^12 + 5010*x^11 - 711*x^10 - 4380*x^9 + 6270*x^8 - 4470*x^7 + 1535*x^6 + 178*x^5 - 450*x^4 + 210*x^3 - 40*x^2 + 1)/(x - 1)^10. E.g.f. for homeomorphically irreducible multigraphs with n nodes and k edges is (1 + x*y)^( - 1/2)*exp(x*y/2 + x^2*y^2/4)*Sum_{k >= 0} 1/(1 - x)^binomial(k, 2)*exp( - x^2*y*k^2/(2*(1 + x*y)) - x^2*y*k/2)*y^k/k!.

A307806 Triangle T(n,k) read by rows: number of series-reduced labeled graphs on n nodes with k components.

Original entry on oeis.org

1, 1, 1, 0, 3, 1, 5, 3, 6, 1, 51, 25, 15, 10, 1, 3634, 381, 90, 45, 15, 1, 374119, 26509, 1596, 280, 105, 21, 1, 73161880, 3095579, 111370, 5061, 770, 210, 28, 1, 26545249985, 671957334, 14411205, 353262, 13671, 1890, 378, 36, 1
Offset: 1

Views

Author

R. J. Mathar, Apr 29 2019

Keywords

Examples

			The triangle starts
1;
1,1;
0,3,1;
5,3,6,1;
51,25,15,10,1;
3634,381,90,45,15,1;
374119,26509,1596,280,105,21,1;
73161880,3095579,111370,5061,770,210,28,1;
26545249985,671957334,14411205,353262,13671,1890,378,36,1;

Crossrefs

Cf. A003515 (column k=1), A003514 (row sums).

Formula

T(n,1) = A003515(n).
T(n,k) = Sum_{Compositions n=n_1+n_2+...n_k, n_i>=1} multinomial(n; n_1,n_2,..,n_k) * T(n_1,1) * T(n_2,1) *... T(n_k,1)/ k!.

A060536 Number of homeomorphically irreducible multigraphs (or series-reduced multigraphs or multigraphs without nodes of degree 2) on 6 labeled nodes.

Original entry on oeis.org

1, 15, 45, 90, 495, 1866, 5990, 19920, 62655, 186525, 526470, 1403265, 3530000, 8388495, 18884475, 40442635, 82775970, 162663240, 308201500, 565176105, 1006419120, 1745321275, 2955037455, 4895398755, 7950135835, 12677752431
Offset: 0

Views

Author

Vladeta Jovovic, Apr 01 2001

Keywords

References

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

Links

Crossrefs

Cf. A003514, A060516, A060533-A060537.

Formula

G.f.: - (6*x^25 - 30*x^24 + 60*x^23 + 615*x^22 - 9280*x^21 + 54909*x^20 - 186150*x^19 + 404285*x^18 - 581340*x^17 + 522915*x^16 - 172878*x^15 - 289605*x^14 + 590880*x^13 - 581955*x^12 + 337755*x^11 - 67650*x^10 - 74150*x^9 + 84315*x^8 - 42870*x^7 + 10410*x^6 + 888*x^5 - 1590*x^4 + 535*x^3 - 75*x^2 + 1)/(x - 1)^15. E.g.f. for homeomorphically irreducible multigraphs with n nodes and k edges is (1 + x*y)^( - 1/2)*exp(x*y/2 + x^2*y^2/4)*Sum_{k >= 0} 1/(1 - x)^binomial(k, 2)*exp( - x^2*y*k^2/(2*(1 + x*y)) - x^2*y*k/2)*y^k/k!.

A060578 Number of homeomorphically irreducible general graphs on 3 labeled node and with n edges.

Original entry on oeis.org

1, 3, 9, 21, 60, 135, 282, 537, 945, 1561, 2451, 3693, 5378, 7611, 10512, 14217, 18879, 24669, 31777, 40413, 50808, 63215, 77910, 95193, 115389, 138849, 165951, 197101, 232734, 273315, 319340, 371337, 429867, 495525, 568941, 650781, 741748
Offset: 0

Views

Author

Vladeta Jovovic, Apr 03 2001

Keywords

Comments

A homeomorphically irreducible general graph is a graph with multiple edges and loops and without nodes of degree 2.

References

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

Links

Crossrefs

Cf. A003514, A060516, A060533-A060537, A060576-A060581.

Programs

  • Mathematica
    CoefficientList[Series[-(8x^9-36x^8+66x^7-70x^6+51x^5-24x^4+8x^3-6x^2+3x-1)/(x-1)^6,{x,0,40}],x] (* Harvey P. Dale, Jul 22 2018 *)

Formula

G.f.: - (8*x^9 - 36*x^8 + 66*x^7 - 70*x^6 + 51*x^5 - 24*x^4 + 8*x^3 - 6*x^2 + 3*x - 1)/(x - 1)^6. E.g.f. for homeomorphically irreducible general graphs with n nodes and k edges is (1 + x*y)^( - 1/2)*exp( - x*y/2 + x^2*y^2/4)*Sum_{k >= 0} 1/(1 - x)^binomial(k + 1, 2)*exp( - x^2*y*k^2/(2*(1 + x*y)) - x^2*y*k/2)*y^k/k!.

A060579 Number of homeomorphically irreducible general graphs on 4 labeled nodes and with n edges.

Original entry on oeis.org

1, 6, 19, 68, 242, 704, 1981, 5140, 12364, 27614, 57598, 113108, 210812, 375606, 643646, 1066196, 1714445, 2685464, 4109493, 6158768, 9058119, 13097592, 18647371, 26175300, 36267330, 49651242, 67224024, 90083308, 119563302
Offset: 0

Views

Author

Vladeta Jovovic, Apr 03 2001

Keywords

Comments

A homeomorphically irreducible general graph is a graph with multiple edges and loops and without nodes of degree 2.

References

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

Links

Crossrefs

Cf. A003514, A060516, A060533-A060537, A060576-A060581.

Formula

G.f.: (4*x^15 + 5*x^14 - 194*x^13 + 881*x^12 - 2058*x^11 + 3096*x^10 - 3330*x^9 + 2628*x^8 - 1398*x^7 + 359*x^6 + 72*x^5 - 93*x^4 + 28*x^3 + 4*x^2 - 4*x + 1)/(x - 1)^10. E.g.f. for homeomorphically irreducible general graphs with n nodes and k edges is (1 + x*y)^( - 1/2)*exp( - x*y/2 + x^2*y^2/4)*Sum_{k >= 0} 1/(1 - x)^binomial(k + 1, 2)*exp( - x^2*y*k^2/(2*(1 + x*y)) - x^2*y*k/2)*y^k/k!.

A060580 Number of homeomorphically irreducible general graphs on 5 labeled nodes and with n edges.

Original entry on oeis.org

1, 10, 40, 185, 765, 2845, 10220, 33885, 105185, 305465, 830811, 2119875, 5091525, 11565505, 24977315, 51552005, 102175360, 195301015, 361365695, 649360880, 1136438375, 1941722170, 3245874555, 5318438260, 8555568895, 13531506921
Offset: 0

Views

Author

Vladeta Jovovic, Apr 03 2001

Keywords

Comments

A homeomorphically irreducible general graph is a graph with multiple edges and loops and without nodes of degree 2.

References

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

Links

Crossrefs

Cf. A003514, A060516, A060533-A060537, A060576-A060581.

Formula

G.f.: - (5*x^22 - 20*x^21 + 23*x^20 - 815*x^19 + 8110*x^18 - 37255*x^17 + 104890*x^16 - 204469*x^15 + 296720*x^14 - 337455*x^13 + 310150*x^12 - 229885*x^11 + 131054*x^10 - 50485*x^9 + 6490*x^8 + 7255*x^7 - 6730*x^6 + 3242*x^5 - 995*x^4 + 180*x^3 - 5*x^2 - 5*x + 1)/(x - 1)^15. E.g.f. for homeomorphically irreducible general graphs with n nodes and k edges is (1 + x*y)^( - 1/2)*exp( - x*y/2 + x^2*y^2/4)*Sum_{k >= 0} 1/(1 - x)^binomial(k + 1, 2)*exp( - x^2*y*k^2/(2*(1 + x*y)) - x^2*y*k/2)*y^k/k!.
Previous Showing 11-17 of 17 results.

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