OFFSET
0,5
COMMENTS
See table A274391 for the coefficients in exp( T^n(x) ), n>=0, where T^n(x) is the e.g.f. of the n-th row of this table.
LINKS
FORMULA
Let T^n(x) denote the n-th iteration of Euler's tree function T(x), then the coefficients in T^n(x) form the n-th row of this table, and the functions satisfy:
(1) T^n(x) = x * exp( Sum_{i=1..n} T^i(x) ).
(2) T^n(x) = T^(n-1)(x) * exp( T^n(x) ).
(3) T^n(x) = T^(n+1)( x/exp(x) ).
EXAMPLE
This table begins:
1, 0, 0, 0, 0, 0, 0, 0, ...;
1, 2, 9, 64, 625, 7776, 117649, 2097152, ...;
1, 4, 30, 332, 4880, 89742, 1986124, 51471800, ...;
1, 6, 63, 948, 18645, 454158, 13221075, 448434136, ...;
1, 8, 108, 2056, 50680, 1537524, 55494712, 2325685632, ...;
1, 10, 165, 3800, 112625, 4090980, 176238685, 8793850560, ...;
1, 12, 234, 6324, 219000, 9266706, 463975764, 26858490392, ...;
1, 14, 315, 9772, 387205, 18704322, 1067280319, 70311813880, ...;
1, 16, 408, 14288, 637520, 34617288, 2217367600, 163802295616, ...;
1, 18, 513, 20016, 993105, 59879304, 4254311817, 348285415872, ...;
1, 20, 630, 27100, 1480000, 98110710, 7656893020, 688058734520, ...;
...
where the e.g.f.s of the rows are iterations of T(x) and begin:
T^0(x) = x;
T^1(x) = T(x) = x + 2*x^2/2! + 9*x^3/3! + 64*x^4/4! + 625*x^5/5! + 7776*x^6/6! + 117649*x^7/7! + 2097152*x^8/8! +...+ n^(n-1)*x^n/n! +...;
T^2(x) = T(T(x)) = x + 4*x^2/2! + 30*x^3/3! + 332*x^4/4! + 4880*x^5/5! + 89742*x^6/6! + 1986124*x^7/7! + 51471800*x^8/8! +...+ A207833(n)*x^n/n! +...;
T^3(x) = T(T(T(x))) = x + 6*x^2/2! + 63*x^3/3! + 948*x^4/4! + 18645*x^5/5! + 454158*x^6/6! + 13221075*x^7/7! + 448434136*x^8/8! +...+ A227278(n)*x^n/n! +...;
T^4(x) = T(T(T(T(x)))) = x + 8*x^2/2! + 108*x^3/3! + 2056*x^4/4! + 50680*x^5/5! + 1537524*x^6/6! + 55494712*x^7/7! + 2325685632*x^8/8! +...;
...
where T^n(x)/exp( T^n(x) ) = T^n( x/exp(x) ) = T^(n-1)(x).
Also we have
T(x) = x*exp( T(x) );
T^2(x) = x*exp( T(x) + T^2(x) );
T^3(x) = x*exp( T(x) + T^2(x) + T^3(x) );
T^4(x) = x*exp( T(x) + T^2(x) + T^3(x) + T^4(x) ); ...
PROG
(PARI) {ITERATE(F, n, k) = my(G=x +x*O(x^k)); for(i=1, n, G=subst(G, x, F)); G}
{T(n, k) = my(TREE = serreverse(x*exp(-x +x*O(x^k)))); k!*polcoeff(ITERATE(TREE, n, k), k)}
/* Print this table as a square array */
for(n=0, 10, for(k=1, 10, print1(T(n, k), ", ")); print(""))
/* Print this table as a flattened array */
for(n=0, 12, for(k=1, n, print1(T(n-k, k), ", ")); )
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Jun 19 2016
STATUS
approved