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A140945 Triangle read by rows: counts series-parallel networks by the number of series connections. 5
1, 1, 1, 1, 6, 1, 1, 25, 25, 1, 1, 90, 290, 90, 1, 1, 301, 2450, 2450, 301, 1, 1, 966, 17451, 41580, 17451, 966, 1, 1, 3025, 112035, 544971, 544971, 112035, 3025, 1, 1, 9330, 671980, 6076350, 12122502, 6076350, 671980, 9330, 1, 1, 28501, 3846700, 60738700, 217523922, 217523922, 60738700, 3846700, 28501, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,5
COMMENTS
T(n,k) is the number of series-parallel matroids on [n+1] of rank k. - Andrew Howroyd, Mar 08 2023
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50; first 17 rows from Brian Drake, Jul 24 2008)
Brian Drake, An inversion theorem for labeled trees and some limits of areas under lattice paths (Example 1.5.1), A dissertation presented to the Faculty of the Graduate School of Arts and Sciences of Brandeis University.
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.
FORMULA
E.g.f. is reversion of log(1+ax)/a+log(1+bx)/b-x.
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
EXAMPLE
Triangle begins:
1;
1, 1;
1, 6, 1;
1, 25, 25, 1;
1, 90, 290, 90, 1;
1, 301, 2450, 2450, 301, 1;
1, 966, 17451, 41580, 17451, 966, 1;
...
MAPLE
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);
PROG
(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))))]}
{ my(A=T(10)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Mar 08 2023
CROSSREFS
Row sums are A006351.
Second column is A000392.
Cf. A359985.
Sequence in context: A174045 A169660 A035348 * A141688 A166960 A155908
KEYWORD
easy,nonn,tabl
AUTHOR
Brian Drake, Jul 24 2008
STATUS
approved

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Last modified July 21 09:38 EDT 2024. Contains 374472 sequences. (Running on oeis4.)