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 1-5 of 5 results.

A136281 Number of graphs on n labeled nodes with degree at most 2.

Original entry on oeis.org

1, 1, 2, 8, 41, 253, 1858, 15796, 152219, 1638323, 19467494, 252998224, 3568259503, 54263159347, 884834059454, 15397757661092, 284767413357977, 5576696746139689, 115269732256964626, 2507575465491619672, 57262481225957071721, 1369461739453440893261
Offset: 0

Views

Author

Don Knuth, Mar 31 2008

Keywords

Comments

These are thunderstorm graphs. Their connected components are a single cycle (clouds), a path (lightning bolts) or an isolated vertex (raindrops). - Geoffrey Critzer, May 11 2011

References

  • D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4.

Links

Crossrefs

Cf. A000085 (degree at most 1), A136282-A136286.

Programs

  • Mathematica
    f = (Log[1/(1-x)]+1/(1-x) -x^2/2 - 1)/2;
Range[0,25]! CoefficientList[Series[Exp[f],{x,0,25}],x] (* Geoffrey Critzer, May 11 2011 *)

Formula

Binomial transform of A000986. E.g.f.: (1-x)^(-1/2)*exp(-x^2/4 + x/((2*(1-x)))). - Vladeta Jovovic, May 20 2008
a(n) = (2*n-1)*a(n-1) - (n-1)^2*a(n-2) + (n-2)*(n-1)*a(n-3) - (n-3)*(n-2)*(n-1)/2*a(n-4). - Vaclav Kotesovec, Aug 10 2013
a(n) ~ n^n*exp(sqrt(2*n)-1/2-n)/sqrt(2) * (1+19/(24*sqrt(2*n))). - Vaclav Kotesovec, Aug 10 2013

Extensions

More terms from Vladeta Jovovic, May 20 2008
a(0)=1 prepended by Alois P. Heinz, Jul 21 2021

A136284 Number of graphs on n labeled nodes with maximal degree exactly 2.

Original entry on oeis.org

0, 0, 4, 31, 227, 1782, 15564, 151455, 1635703, 19457998, 252962528, 3568119351, 54262590843, 884831668974, 15397747311556, 284767367151241, 5576696534340377, 115269731259650802, 2507575460681918232, 57262481202198407625, 1369461739333488200365
Offset: 1

Views

Author

Don Knuth, Mar 31 2008

Keywords

References

  • D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4.

Links

Crossrefs

Cf. A000085 (degree at most 1), A136281, A136282, A136283, A136285, A136286.

Programs

  • Mathematica
    nn = 20; Drop[Range[0, nn]! CoefficientList[Series[Exp[1/(1 - z)/2 - 1/2 + Log[1/(1 - z)]/2 - z^2/4] - Exp[z + z^2/2!], {z, 0, nn}], z], 1] (* Geoffrey Critzer, Jul 23 2016 *)
  • PARI
    x='x+O('x^22); concat( [0,0], Vec( serlaplace( exp(1/(1-x)/2 - 1/2 + log(1/(1-x))/2-x^2/4) - exp(x+x^2/2!) ) ) ) \\ Joerg Arndt, Jul 24 2016

Formula

Equals A136281 - A000085.
Recurrence: 2*(n-3)*(9*n-64)*a(n) = 2*(18*n^3 - 182*n^2 + 423*n - 149)*a(n-1) - 2*(n-1)*(9*n^3 - 91*n^2 + 243*n - 173)*a(n-2) + 6*(n-2)*(n-1)*(n+1)*a(n-3) + (n-3)*(n-2)*(n-1)*(9*n^2 - 91*n + 224)*a(n-4) - (n-4)*(n-3)*(n-2)*(n-1)*(9*n-67)*a(n-5) + (n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(9*n-55)*a(n-6). - Vaclav Kotesovec, Feb 09 2014
a(n) ~ exp(sqrt(2*n)-n-1/2) * n^n / sqrt(2) * (1 + 19/(24*sqrt(2*n))). - Vaclav Kotesovec, Feb 09 2014
E.g.f.: exp(1/(1-x)/2 - 1/2 + log(1/(1-x))/2-x^2/4) - exp(x+x^2/2!). - Joerg Arndt, Jul 24 2016

Extensions

More terms from Alois P. Heinz, Sep 12 2008

A136282 Number of graphs on n labeled nodes with degree at most 3.

Original entry on oeis.org

1, 2, 8, 64, 768, 12068, 236926, 5651384, 160054952, 5284391984, 200375581984, 8620342917808, 416471882713712, 22400989824444576, 1331457489258580672, 86887134810544955072, 6189888588922841477824, 478992737680928902742656, 40082045451011806706919808, 3612470757307682016196841216, 349398857659776033845292636416
Offset: 1

Views

Author

Don Knuth, Mar 31 2008

Keywords

References

  • D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4.

Links

Crossrefs

Cf. A000085 (degree at most 1), A136281-A136286.

Formula

Binomial transform of A110041. - Vladeta Jovovic, May 20 2008
Recurrence: 12*(81*n^4 - 837*n^3 + 3375*n^2 - 6171*n + 4192)*a(n) = 6*(243*n^5 - 2511*n^4 + 10665*n^3 - 21969*n^2 + 19476*n - 4624)*a(n-1) + 3*(n-1)*(243*n^6 - 2997*n^5 + 15147*n^4 - 39843*n^3 + 57594*n^2 - 41832*n + 10888)*a(n-2) - 3*(n-2)*(n-1)*(405*n^5 - 3699*n^4 + 13527*n^3 - 22629*n^2 + 14048*n + 388)*a(n-3) + (n-3)*(n-2)*(n-1)*(243*n^5 - 1944*n^4 + 6777*n^3 - 9738*n^2 - 2370*n + 10732)*a(n-4) + 2*(n-4)*(n-3)*(n-2)*(n-1)*(81*n^4 - 999*n^3 + 4968*n^2 - 8646*n + 4906)*a(n-5) + (n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(243*n^5 - 2916*n^4 + 12933*n^3 - 27990*n^2 + 27978*n - 8948)*a(n-6) - (n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(81*n^4 - 513*n^3 + 891*n^2 - 357*n - 242)*a(n-7) - (n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(81*n^4 - 513*n^3 + 1350*n^2 - 1608*n + 640)*a(n-8). - Vaclav Kotesovec, Aug 13 2013
a(n) ~ 3^(n/2) * exp(sqrt(3*n) - 3*n/2 - 5/4) * n^(3*n/2) / 2^(n + 1/2) * (1 + 71/(24*sqrt(3*n))). - Vaclav Kotesovec, Nov 05 2023
a(n) / A110041(n) ~ 1 + 2/sqrt(3*n). - Vaclav Kotesovec, Nov 06 2023

Extensions

More terms from Vladeta Jovovic, May 20 2008

A136283 Number of graphs on n labeled nodes with degree at most 4.

Original entry on oeis.org

1, 2, 8, 64, 1024, 27449, 1052793, 53470067, 3451287371, 275322712826, 26566919914276, 3047264283807562, 409523561958024458, 63703577287372238069, 11351386036074641226649, 2296295762734645223170899, 523223196906671550193022083, 133357299601279100522308107142
Offset: 1

Views

Author

Don Knuth, Mar 31 2008

Keywords

References

  • D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4.

Links

Crossrefs

Cf. A000085 (degree at most 1), A136281-A136286.

Programs

  • PARI
    GraphsWithDegreeAtMost(n,limit)={
local(M=Map());
my(acc(p,v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));
my(recurse(p,i,q,v,e)=if(e<=limit&&poldegree(q)<=limit, if(i<0, acc(x^e+q, v), my(t=polcoeff(p,i)); for(k=0, t, self()(p, i-1, (t-k+x*k)*x^i+q, binomial(t,k)*v, e+k)))));
my(iterate(v,k,f)=for(i=1,k,v=f(v));v);
vecsum(Vec(iterate(Mat([1,1]), n-1, src->M=Map(); for(i=1, matsize(src)[1], my(p=src[i,1]); recurse(p,poldegree(p),0,src[i,2],0)); Mat(M)))[2]); }
a(n) = GraphsWithDegreeAtMost(n, 4); \\ Andrew Howroyd, Aug 25 2017

Extensions

Terms a(13) and beyond from Andrew Howroyd, Aug 25 2017

A136285 Number of graphs on n labeled nodes with maximal degree exactly 3.

Original entry on oeis.org

0, 0, 0, 23, 515, 10210, 221130, 5499165, 158416629, 5264924490, 200122583760, 8616774658305, 416417619554365, 22400104990385122, 1331442091500919580, 86886850043131597095, 6189883012226095338135, 478992622411196645778030
Offset: 1

Views

Author

Don Knuth, Mar 31 2008

Keywords

References

  • D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4.

Links

Crossrefs

Cf. A000085 (degree at most 1), A136281-A136286.

Formula

Equals A136282 - A136281.

Extensions

More terms from Alois P. Heinz, Sep 12 2008
Showing 1-5 of 5 results.

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