A361355 -id:A361355 - cp's OEIS frontend

A007834 Number of point labeled reduced 5-free two-graphs with n nodes.

1, 0, 1, 1, 16, 76, 1016, 10284, 157340, 2411756, 44953712, 899824256, 20283419872, 495216726096, 13202082981712, 378896535199888, 11690436112988224, 385173160930360192, 13509981115738946816, 502374681770910293568, 19746124320077115154112, 817908018939079281840320

Offset: 1

Peter J. Cameron

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..200
P. J. Cameron, Counting two-graphs related to trees, Elec. J. Combin., Vol. 2, #R4.

Row sums are A361355.

Cf. A007831, A007832, A007833, A359986.

CoefficientList[Series[-2*LambertW[-1/2*E^(-1/2)*(1+x)^(1/2)]/(1+x), {x, 0, 15}], x]* Range[0, 15]! (* Vaclav Kotesovec, Sep 30 2013 *)

\\ B(x) gives the e.g.f. of A359986.
B(n)={exp(2*x + intformal(serreverse(log(1 + x + O(x^n)) + log(1 + x + O(x^n)) - x)))}
seq(n)={Vec(serlaplace(log(subst(B(n), x, log(1 + x + O(x*x^n)))/(1 + x))))} \\ Andrew Howroyd, Oct 15 2024

E.g.f.: -2*LambertW(-1/2*exp(-1/2)*(1+x)^(1/2))/(1+x). - Vladeta Jovovic, Aug 21 2006
a(n) ~ sqrt(2)*sqrt(4-exp(1)) * n^(n-1) / (8*exp(n-1)*(4*exp(-1)-1)^n). - Vaclav Kotesovec, Sep 30 2013
E.g.f.: log(B(log(1 + x))/(1 + x)), where B(x) is the e.g.f. of A359986. - Andrew Howroyd, Oct 15 2024

a(20) onwards from Andrew Howroyd, Oct 15 2024

A361353 Triangle read by rows: T(n,k) is the number of simple quasi series-parallel matroids on [n] with rank k, 1 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 5, 1, 0, 0, 15, 16, 1, 0, 0, 0, 175, 42, 1, 0, 0, 0, 735, 1225, 99, 1, 0, 0, 0, 0, 16065, 6769, 219, 1, 0, 0, 0, 0, 76545, 204400, 32830, 466, 1, 0, 0, 0, 0, 0, 2747745, 2001230, 147466, 968, 1, 0, 0, 0, 0, 0, 13835745, 56143395, 16813720, 632434, 1981, 1

Offset: 1

Andrew Howroyd, Mar 09 2023

See Table 2 in the Ferroni/Larson reference.

			Triangle begins:
  1;
  0, 1;
  0, 1,  1;
  0, 0,  5,   1;
  0, 0, 15,  16,     1;
  0, 0,  0, 175,    42,      1;
  0, 0,  0, 735,  1225,     99,     1;
  0, 0,  0,   0, 16065,   6769,   219,   1;
  0, 0,  0,   0, 76545, 204400, 32830, 466, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
Luis Ferroni and Matt Larson, Kazhdan-Lusztig polynomials of braid matroids, arXiv:2303.02253 [math.CO], 2023.
Nicholas Proudfoot, Yuan Xu, and Ben Young, On the enumeration of series-parallel matroids, arXiv:2406.04502 [math.CO], 2024.

Row sums are A361354.

Cf. A140945, A359985, A361355.

\\ B gives A359985 as e.g.f.
B(n)= {exp(x*(1+y) + y*intformal(serreverse(log(1 + x*y + O(x^n))/y + log(1 + x + O(x^n)) - x)))}
T(n) = {[Vecrev(p/y) | p<-Vec(serlaplace(subst(B(n), x, log(1 + x + O(x*x^n)))/(1 + x) - 1))]}
{ my(A=T(9)); for(i=1, #A, print(A[i])) }

E.g.f.: A(x,y) = B(log(1 + x), y)/(1 + x) - 1 where B(x,y) is the e.g.f. of A359985.

A361282 Number of rank n+1 simple connected series-parallel matroids on [2n].

Original entry on oeis.org

0, 1, 75, 9345, 1865745, 554479695, 231052877055, 128938132548225, 92986310399407425, 84250567868935042575, 93744545254140599193375, 125717783386887888296925825, 200041202339679732328342670625, 372688996228146502285257581079375, 803768398459351988653830600415029375

Offset: 1

Matt Larson, Mar 06 2023

nonn

			For n=2 the a(2) = 1 rank 3 simple connected series-parallel matroid on [4] is the uniform matroid of rank 3.

Luis Ferroni and Matt Larson, Kazhdan-Lusztig polynomials of braid matroids, arXiv:2303.02253 [math.CO], 2023.

Cf. A034941, A361355.

a(n) = T(2*n)[2*n][n+1] \\ T(n) defined in A361355. - Andrew Howroyd, Mar 09 2023

Terms a(10)-a(15) from Andrew Howroyd, Mar 09 2023

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A007834 Number of point labeled reduced 5-free two-graphs with n nodes.

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A361353 Triangle read by rows: T(n,k) is the number of simple quasi series-parallel matroids on [n] with rank k, 1 <= k <= n.

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A361282 Number of rank n+1 simple connected series-parallel matroids on [2n].

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