A274391 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.
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, 1, 11, 133, 1345, 8321, 16807, 1, 1, 1, 13, 196, 2729, 28396, 162463, 262144, 1, 1, 1, 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
Offset: 0
Examples
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.
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Programs
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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),", "));)
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).
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