A331437 - OEIS

%I #20 Jan 24 2020 19:16:11

%S 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,

%T 20,180,1260,6448,0,0,0,0,15,420,3780,23520,56961,0,0,0,0,10,700,

%U 10850,79800,347760,892720,0,0,0,0,1,837,24045,269360,1655640,6400800,11905091

%N 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.

%C Homeomorphically irreducible graphs are graphs without vertices of degree 2. - _Andrew Howroyd_, Jan 24 2020

%H Andrew Howroyd, <a href="/A331437/b331437.txt">Table of n, a(n) for n = 0..1325</a> (rows n = 0..50)

%H D. M. Jackson and J. W. Reilly, <a href="https://doi.org/10.1016/0095-8956(75)90090-8">The enumeration of homeomorphically irreducible labeled graphs</a>, J. Combin. Theory, B 19 (1975), 272-286. See Table III.

%e Triangle begins:

%e   1;

%e   0, 1;

%e   0, 0, 0;

%e   0, 0, 0, 4;

%e   0, 0, 0, 0,  5;

%e   0, 0, 0, 0,  0,  96;

%e   0, 0, 0, 1,  0, 120,  427;

%e   0, 0, 0, 0, 20, 180, 1260,  6448;

%e   0, 0, 0, 0, 15, 420, 3780, 23520, 56961;

%e ...

%o (PARI) \\ See Jackson & Reilly for e.g.f.

%o 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!)}

%o T(n) = {Mat([Col(p, -n) | p<-Vec(serlaplace(log(H(n,y + O(y^n)))))])}

%o { my(A=T(10)); for(n=1, #A, print(A[n, 1..n])) } \\ _Andrew Howroyd_, Jan 24 2020

%Y Column sums are A003515.

%Y Row sums are A331584.

%Y Right diagonal is A005512(n+1).

%Y Cf. A060514, A331438 (transpose).

%K nonn,tabl

%O 0,10

%A _N. J. A. Sloane_, Jan 19 2020

%E Terms a(44) and beyond from _Andrew Howroyd_, Jan 24 2020