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A140945
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Triangle read by rows: counts series-parallel networks by the number of series connections.
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5
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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
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OFFSET
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1,5
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COMMENTS
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T(n,k) is the number of series-parallel matroids on [n+1] of rank k. - Andrew Howroyd, Mar 08 2023
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LINKS
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FORMULA
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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
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EXAMPLE
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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;
...
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MAPLE
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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);
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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