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A331437 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. 6
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,10

COMMENTS

Homeomorphically irreducible graphs are graphs without vertices of degree 2. - Andrew Howroyd, Jan 24 2020

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows n = 0..50)

D. M. Jackson and J. W. Reilly, The enumeration of homeomorphically irreducible labeled graphs, J. Combin. Theory, B 19 (1975), 272-286. See Table III.

EXAMPLE

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;

...

PROG

(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

CROSSREFS

Column sums are A003515.

Row sums are A331584.

Right diagonal is A005512(n+1).

Cf. A060514, A331438 (transpose).

Sequence in context: A323519 A333979 A276580 * A108708 A290322 A274948

Adjacent sequences:  A331434 A331435 A331436 * A331438 A331439 A331440

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, Jan 19 2020

EXTENSIONS

Terms a(44) and beyond from Andrew Howroyd, Jan 24 2020

STATUS

approved

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Last modified July 31 17:51 EDT 2021. Contains 346376 sequences. (Running on oeis4.)