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
Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
Shi-Mei Ma, Some combinatorial sequences associated with context-free grammars, arXiv:1208.3104v2 [math.CO], 2012. - N. J. A. Sloane, Aug 21 2012
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
KEYWORD
nonn,tabl
AUTHOR
Christian G. Bower, May 15 2000
STATUS
approved