login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A055356
Triangle of increasing mobiles (circular rooted trees) with n nodes and k leaves.
12
1, 1, 0, 1, 1, 0, 1, 4, 2, 0, 1, 11, 18, 6, 0, 1, 26, 98, 96, 24, 0, 1, 57, 424, 874, 600, 120, 0, 1, 120, 1614, 6040, 8244, 4320, 720, 0, 1, 247, 5682, 35458, 83500, 83628, 35280, 5040, 0, 1, 502, 19022, 187288, 701164, 1169768, 915984, 322560, 40320, 0, 1
OFFSET
1,8
COMMENTS
In an increasing rooted tree, nodes are numbered and numbers increase as you move away from root.
Also related to the solution of the equation df/dt=f e^f (see the Maple code). - F. Chapoton, Jul 16 2004
LINKS
FORMULA
Let p(n,x) be the polynomial with coefficients equal to the n-th row of the triangle in ascending powers of x, e.g., p(4,x) = 1+4*x+2*x^2; then p(n+1,x) = (1+(n-1)*x)*p(n,x) + x*p'(n,x). - Ben Whitmore, May 12 2021
Recurrence: T(n,k) = (n-2) * T(n-1,k-1) + k * T(n-1,k) for n >= 1, 1 <= k <= n with T(1,1) = 1 and T(n,k) = 0 for n < 1, k < 1 or k > n. - Georg Fischer, Oct 27 2021
EXAMPLE
Triangle begins
1;
1, 0;
1, 1, 0;
1, 4, 2, 0;
1, 11, 18, 6, 0;
1, 26, 98, 96, 24, 0;
1, 57, 424, 874, 600, 120, 0;
...
MAPLE
P[1]:=1; for n from 1 to 8 do P[n+1]:=simplify((1+n*x)*P[n]+x*diff(P[n], x)) end; # F. Chapoton, Jul 16 2004
MATHEMATICA
P[1][_] = 1;
P[n_][x_] := P[n][x] = (1 + (n-1) x) P[n-1][x] + x P[n-1]'[x] // Expand;
row[1] = {1};
row[n_] := Append[CoefficientList[P[n-1][x], x], 0];
Array[row, 10] // Flatten (* Jean-François Alcover, Nov 17 2018, after F. Chapoton *)
PROG
(PARI)
A(n)={my(v=vector(n)); v[1]=y; for(n=2, #v, v[n]=v[n-1] + sum(k=1, n-2, binomial(n-2, k)*v[k]*v[n-k])); vector(#v, i, Vecrev(v[i]/y, i))}
{ my(T=A(10)); for(i=1, #T, print(T[i])) } \\ Andrew Howroyd, Sep 23 2018
CROSSREFS
Row sums give A029768 (p(n,1)).
Alternating row sums give A089963 (p(n+1,-1)).
Sequence in context: A121225 A216715 A049430 * A297331 A028956 A129681
KEYWORD
nonn,tabl
AUTHOR
Christian G. Bower, May 15 2000
STATUS
approved

  NODES
orte 1
see 2
Story 1