A339285 Triangle read by rows: T(n,k) is the number of unoriented series-parallel networks whose multigraph has n edges and k interior vertices, 0 <= k < n.

1, 1, 1, 1, 2, 1, 1, 4, 5, 1, 1, 6, 14, 8, 1, 1, 9, 34, 39, 14, 1, 1, 12, 68, 132, 94, 20, 1, 1, 16, 126, 370, 447, 202, 30, 1, 1, 20, 212, 887, 1625, 1275, 398, 40, 1, 1, 25, 340, 1911, 4955, 5985, 3284, 730, 55, 1, 1, 30, 515, 3765, 13133, 22608, 19245, 7649, 1266, 70, 1

Offset

1,5

Comments

Unoriented version of A339231. Equivalence is up to reversal of all parts combined in series.

Links

Table of n, a(n) for n=1..66.

Formula

T(n,0) = T(n,n-1) = 1.

T(n,1) = A002620(n).

A339286(n) = Sum_{k=1..n-1} k*T(n,k).

Example

Triangle begins:
  1;
  1,  1;
  1,  2,   1;
  1,  4,   5,    1;
  1,  6,  14,    8,    1;
  1,  9,  34,   39,   14,    1;
  1, 12,  68,  132,   94,   20,    1;
  1, 16, 126,  370,  447,  202,   30,   1;
  1, 20, 212,  887, 1625, 1275,  398,  40,  1;
  1, 25, 340, 1911, 4955, 5985, 3284, 730, 55, 1;
  ...
T(4,0) = 1: (o|o|o|o).
T(4,1) = 4: ((o|o)(o|o)), (o(o|o|o)), (o|o|oo), (o|o(o|o)).
T(4,2) = 5: (oo(o|o)), (o(o|o)o),  (o(o|oo)),  (oo|oo), (o|ooo).
T(4,3) = 1: (oooo).

Prog

(PARI)
EulerMT(u)={my(n=#u, p=x*Ser(u), vars=variables(p)); Vec(exp( sum(i=1, n, substvec(p + O(x*x^(n\i)), vars, [v^i|v<-vars])/i ))-1)}
SubPwr(p, e)={my(vars=variables(p)); substvec(p, vars, [v^e|v<-vars])}
BW(n, Z, W)={my(p=Z+O(x^2)); for(n=2, n, p=x*Ser(EulerMT(Vec(W*p^2/(1+W*p)+Z)))); p}
VertexWeighted(n, Z, W)={my(q=SubPwr(BW((n+1)\2, Z, W), 2), W2=SubPwr(W, 2), s=SubPwr(Z, 2)+W2*q^2/(1+W2*q), p=Z+O(x^2), t=p); for(n=1, n\2, t=Z + q*(W + W2*p); p=Z + x*Ser(EulerMT(Vec(t+(s-SubPwr(t, 2))/2))) - t); Vec(p+t-Z+BW(n, Z, W))/2}
T(n)={[Vecrev(p)|p<-VertexWeighted(n, x, y)]}
{ my(A=T(12)); for(n=1, #A, print(A[n])) }

Crossrefs

Row sums are A339225.

Cf. A002620, A339231, A339282, A339286.

Sequence in context: A197380 A057785 A305882 * A363043 A192404 A291261

Adjacent sequences: A339282 A339283 A339284 * A339286 A339287 A339288

Keyword

nonn,tabl

Author

Andrew Howroyd, Nov 30 2020

Status

approved