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

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

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)

B. 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.

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

Adjacent sequences: A140942 A140943 A140944 * A140946 A140947 A140948

Keyword

easy,nonn,tabl

Author

Brian Drake, Jul 24 2008

Status

approved