OFFSET
0,9
COMMENTS
The e.g.f. of each row is an infinite exponential tetration of the e.g.f. of the prior row: W_{n+1}(x) = W_n(x)^W_n(x)^W_n(x)^..., starting with exp(x) as the e.g.f. of row zero. All of these row functions may be expressed in terms of the LambertW(x) function.
LINKS
FORMULA
Let W_n(x) denote the e.g.f. of the n-th row function of this table, and T^n(x) the n-th iteration of Euler's tree function T(x) (cf. A274390), then
(1) W_n(x) = exp( T^n(x) ).
(2) W_n(x) = T^n(x) / T^(n-1)(x).
(3) W_n(x) = W_{n+1}( x/exp(x) ).
(4) W_n(x) = W_n( x/exp(x) )^W_n(x).
EXAMPLE
This table begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, ...;
1, 1, 3, 16, 125, 1296, 16807, 262144, 4782969, ...;
1, 1, 5, 43, 525, 8321, 162463, 3774513, 101808185, ...;
1, 1, 7, 82, 1345, 28396, 734149, 22485898, 796769201, ...;
1, 1, 9, 133, 2729, 71721, 2300485, 87194689, 3815719969, ...;
1, 1, 11, 196, 4821, 151376, 5787931, 261066156, 13577077401, ...;
1, 1, 13, 271, 7765, 283321, 12567187, 656778529, 39536713209, ...;
1, 1, 15, 358, 11705, 486396, 24539593, 1457297878, 99609347825, ...;
1, 1, 17, 457, 16785, 782321, 44223529, 2940281793, 224869459201, ...;
1, 1, 19, 568, 23149, 1195696, 74840815, 5506111864, 465734919289, ...;
1, 1, 21, 691, 30941, 1754001, 120403111, 9709554961, 899836571001, ...;
...
in which the e.g.f. of row n equals W_n(x) = exp( T^n(x) ), where T^n(x) is the n-th iteration of the Euler tree function T(x).
The row functions begin:
W_0(x) = 1 + x + x^2/2! + x^3/3! + x^4/4! + x^5/5! + x^6/6! +...;
W_1(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 125*x^4/4! + 1296*x^5/5! + 16807*x^6/6! + +...+ (n+1)^(n-1)*x^n/n! +...;
W_2(x) = 1 + x + 5*x^2/2! + 43*x^3/3! + 525*x^4/4! + 8321*x^5/5! + 162463*x^6/6! + +...+ A227176(n)*x^n/n! +...;
W_3(x) = 1 + x + 7*x^2/2! + 82*x^3/3! + 1345*x^4/4! + 28396*x^5/5! + 734149*x^6/6! +...+ A268653(n)*x^n/n! +...;
W_4(x) = 1 + x + 9*x^2/2! + 133*x^3/3! + 2729*x^4/4! + 71721*x^5/5! + 2300485*x^6/6! +...+ A268654(n)*x^n/n! +...;
W_5(x) = 1 + x + 11*x^2/2! + 196*x^3/3! + 4821*x^4/4! + 151376*x^5/5! + 5787931*x^6/6! +...;
W_6(x) = 1 + x + 13*x^2/2! + 271*x^3/3! + 7765*x^4/4! + 283321*x^5/5! + 12567187*x^6/6! +...;
...
and satisfy:
(0) W_0(x) = exp(x),
(1) W_1(x) = exp(x)^W_1(x) = exp(T(x)) = LambertW(-x)/(-x),
(2) W_2(x) = W_1(x)^W_2(x) = exp(T(T(x))),
(3) W_3(x) = W_2(x)^W_3(x) = exp(T(T(T(x)))),
(4) W_4(x) = W_3(x)^W_4(x) = exp(T(T(T(T(x))))),
...
Euler's tree function T(x), and its iterates begin:
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(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(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(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! +...
...
Note that the e.g.f. of the n-th row function, W_n(x), also equals the ratio of two iterates of the Euler tree function: W_n(x) = T^n(x) / T^(n-1)(x).
See A274390 for the table of coefficients in these iterated tree functions.
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(exp(ITERATE(TREE, n, k)), k)}
/* Print this table as a square array */
for(n=0, 10, for(k=0, 10, print1(T(n, k), ", ")); print(""))
/* Print this table as a flattened array */
for(n=0, 12, for(k=0, n, print1(T(n-k, k), ", ")); )
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Jun 19 2016
STATUS
approved