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A140945 Triangle read by rows: counts series-parallel networks by the number of series connections.

%I A140945 #25 Jun 14 2024 04:11:36
%S A140945 1,1,1,1,6,1,1,25,25,1,1,90,290,90,1,1,301,2450,2450,301,1,1,966,
%T A140945 17451,41580,17451,966,1,1,3025,112035,544971,544971,112035,3025,1,1,
%U A140945 9330,671980,6076350,12122502,6076350,671980,9330,1,1,28501,3846700,60738700,217523922,217523922,60738700,3846700,28501,1
%N A140945 Triangle read by rows: counts series-parallel networks by the number of series connections.
%C A140945 T(n,k) is the number of series-parallel matroids on [n+1] of rank k. - _Andrew Howroyd_, Mar 08 2023
%H A140945 Andrew Howroyd, <a href="/A140945/b140945.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50; first 17 rows from Brian Drake, Jul 24 2008)
%H A140945 Brian Drake, <a href="http://people.brandeis.edu/~gessel/homepage/students/drakethesis.pdf">An inversion theorem for labeled trees and some limits of areas under lattice paths (Example 1.5.1)</a>, A dissertation presented to the Faculty of the Graduate School of Arts and Sciences of Brandeis University.
%H A140945 Luis Ferroni and Matt Larson, <a href="https://arxiv.org/abs/2303.02253">Kazhdan-Lusztig polynomials of braid matroids</a>, arXiv:2303.02253 [math.CO], 2023.
%H A140945 Nicholas Proudfoot, Yuan Xu, and Ben Young, <a href="https://arxiv.org/abs/2406.04502">On the enumeration of series-parallel matroids</a>, arXiv:2406.04502 [math.CO], 2024.
%F A140945 E.g.f. is reversion of log(1+ax)/a+log(1+bx)/b-x.
%F A140945 Let f(x,t) = (1+x)*(1+x*t)/(1-x^2*t) and let D be the operator f(x,t)*d/dx. Then the n-th row polynomial equals (D^n)(f(x,t)) evaluated at x = 0. - Peter Bala, Sep 29 2011
%e A140945 Triangle begins:
%e A140945   1;
%e A140945   1,   1;
%e A140945   1,   6,     1;
%e A140945   1,  25,    25,     1;
%e A140945   1,  90,   290,    90,     1;
%e A140945   1, 301,  2450,  2450,   301,   1;
%e A140945   1, 966, 17451, 41580, 17451, 966, 1;
%e A140945   ...
%p A140945 N:=6: 1/a*log(1+a*y)+1*log(1+b*y)/b-y=x: solve(%, y):series(%, x, N): simplify(%, symbolic): convert(%, polynom): subs(b=1, %): R:= [seq(i!*coeff(%, x, i), i=1..N-1)]: seq( seq(coeff(R[i], a, j), j=0..i-1), i=1..N-1);
%o A140945 (PARI) T(n) = {[Vecrev(p) | p<-Vec(serlaplace(intformal(serreverse(log(1 + x*y + O(x*x^n))/y + log(1 + x + O(x*x^n)) - x))))]}
%o A140945 { my(A=T(10)); for(i=1, #A, print(A[i])) }  \\ _Andrew Howroyd_, Mar 08 2023
%Y A140945 Row sums are A006351.
%Y A140945 Second column is A000392.
%Y A140945 Cf. A359985.
%K A140945 easy,nonn,tabl
%O A140945 1,5
%A A140945 _Brian Drake_, Jul 24 2008