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A331437
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Triangle read by rows: T(n,k) = number of homeomorphically irreducible connected labeled graphs with n edges and k vertices, n >= 0, 1 <= k <= n+1.
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6
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1, 0, 1, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 96, 0, 0, 0, 1, 0, 120, 427, 0, 0, 0, 0, 20, 180, 1260, 6448, 0, 0, 0, 0, 15, 420, 3780, 23520, 56961, 0, 0, 0, 0, 10, 700, 10850, 79800, 347760, 892720, 0, 0, 0, 0, 1, 837, 24045, 269360, 1655640, 6400800, 11905091
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OFFSET
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0,10
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COMMENTS
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Homeomorphically irreducible graphs are graphs without vertices of degree 2. - Andrew Howroyd, Jan 24 2020
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LINKS
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EXAMPLE
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Triangle begins:
1;
0, 1;
0, 0, 0;
0, 0, 0, 4;
0, 0, 0, 0, 5;
0, 0, 0, 0, 0, 96;
0, 0, 0, 1, 0, 120, 427;
0, 0, 0, 0, 20, 180, 1260, 6448;
0, 0, 0, 0, 15, 420, 3780, 23520, 56961;
...
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PROG
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(PARI) \\ See Jackson & Reilly for e.g.f.
H(n, y) = {my(A=O(x*x^n)); (exp(y*x/2 - (y*x)^2/4 + A)/sqrt(1 + y*x + A))*sum(k=0, n, ((1 + y)*exp(-y^2*x/(1+y*x) + A))^binomial(k, 2) * (x*exp((y^3*x^2 + A)/(2*(1 + y*x))))^k / k!)}
T(n) = {Mat([Col(p, -n) | p<-Vec(serlaplace(log(H(n, y + O(y^n)))))])}
{ my(A=T(10)); for(n=1, #A, print(A[n, 1..n])) } \\ Andrew Howroyd, Jan 24 2020
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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