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%I #44 Jul 05 2016 13:03:16
%S 1,1,1,1,1,1,1,1,3,1,1,1,5,16,1,1,1,7,43,125,1,1,1,9,82,525,1296,1,1,
%T 1,11,133,1345,8321,16807,1,1,1,13,196,2729,28396,162463,262144,1,1,1,
%U 15,271,4821,71721,734149,3774513,4782969,1,1,1,17,358,7765,151376,2300485,22485898,101808185,100000000,1,1,1,19,457,11705,283321,5787931,87194689,796769201,3129525793,2357947691,1,1,1,21,568,16785,486396,12567187,261066156,3815719969,32084546824,108063152091,61917364224,1,1,1,23,691,23149,782321,24539593,656778529,13577077401,189440927857,1447917011461,4143297446729,1792160394037,1,1,1,25,826,30941,1195696,44223529,1457297878,39536713209,800175234736,10525328121221,72411962077126,174723134310277,56693912375296,1
%N Table of coefficients in functions that satisfy W_n(x) = W_{n-1}(x)^W_n(x), with W_0(x) = exp(x), as read by antidiagonals.
%C 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.
%H Paul D. Hanna, <a href="/A274391/b274391.txt">Table of n, a(n) for n = 0..1080, of rows 0..45 of the flattened table.</a>
%F 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
%F (1) W_n(x) = exp( T^n(x) ).
%F (2) W_n(x) = T^n(x) / T^(n-1)(x).
%F (3) W_n(x) = W_{n+1}( x/exp(x) ).
%F (4) W_n(x) = W_n( x/exp(x) )^W_n(x).
%e This table begins:
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, ...;
%e 1, 1, 3, 16, 125, 1296, 16807, 262144, 4782969, ...;
%e 1, 1, 5, 43, 525, 8321, 162463, 3774513, 101808185, ...;
%e 1, 1, 7, 82, 1345, 28396, 734149, 22485898, 796769201, ...;
%e 1, 1, 9, 133, 2729, 71721, 2300485, 87194689, 3815719969, ...;
%e 1, 1, 11, 196, 4821, 151376, 5787931, 261066156, 13577077401, ...;
%e 1, 1, 13, 271, 7765, 283321, 12567187, 656778529, 39536713209, ...;
%e 1, 1, 15, 358, 11705, 486396, 24539593, 1457297878, 99609347825, ...;
%e 1, 1, 17, 457, 16785, 782321, 44223529, 2940281793, 224869459201, ...;
%e 1, 1, 19, 568, 23149, 1195696, 74840815, 5506111864, 465734919289, ...;
%e 1, 1, 21, 691, 30941, 1754001, 120403111, 9709554961, 899836571001, ...;
%e ...
%e 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).
%e The row functions begin:
%e W_0(x) = 1 + x + x^2/2! + x^3/3! + x^4/4! + x^5/5! + x^6/6! +...;
%e 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! +...;
%e 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! +...;
%e 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! +...;
%e 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! +...;
%e 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! +...;
%e 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! +...;
%e ...
%e and satisfy:
%e (0) W_0(x) = exp(x),
%e (1) W_1(x) = exp(x)^W_1(x) = exp(T(x)) = LambertW(-x)/(-x),
%e (2) W_2(x) = W_1(x)^W_2(x) = exp(T(T(x))),
%e (3) W_3(x) = W_2(x)^W_3(x) = exp(T(T(T(x)))),
%e (4) W_4(x) = W_3(x)^W_4(x) = exp(T(T(T(T(x))))),
%e ...
%e Euler's tree function T(x), and its iterates begin:
%e 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! +...
%e 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! +...
%e 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! +...
%e 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! +...
%e ...
%e 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).
%e See A274390 for the table of coefficients in these iterated tree functions.
%o (PARI) {ITERATE(F,n,k) = my(G=x +x*O(x^k)); for(i=1,n,G=subst(G,x,F));G}
%o {T(n,k) = my(TREE = serreverse(x*exp(-x +x*O(x^k)))); k!*polcoeff(exp(ITERATE(TREE,n,k)),k)}
%o /* Print this table as a square array */
%o for(n=0,10,for(k=0,10,print1(T(n,k),", "));print(""))
%o /* Print this table as a flattened array */
%o for(n=0,12,for(k=0,n,print1(T(n-k,k),", "));)
%Y Cf. A274390, A000272, A227176, A268653, A268654; diagonals: A274387, A274388.
%Y Cf. A274741 (same table, but read differently).
%K nonn,tabl
%O 0,9
%A _Paul D. Hanna_, Jun 19 2016