A274390 Table of coefficients in the iterations of Euler's tree function (A000169), as read by antidiagonals.
1, 1, 0, 1, 2, 0, 1, 4, 9, 0, 1, 6, 30, 64, 0, 1, 8, 63, 332, 625, 0, 1, 10, 108, 948, 4880, 7776, 0, 1, 12, 165, 2056, 18645, 89742, 117649, 0, 1, 14, 234, 3800, 50680, 454158, 1986124, 2097152, 0, 1, 16, 315, 6324, 112625, 1537524, 13221075, 51471800, 43046721, 0, 1, 18, 408, 9772, 219000, 4090980, 55494712, 448434136, 1530489744, 1000000000, 0, 1, 20, 513, 14288, 387205, 9266706, 176238685, 2325685632, 17386204761, 51395228090, 25937424601, 0, 1, 22, 630, 20016, 637520, 18704322, 463975764, 8793850560, 111107380464, 759123121050, 1924687118684, 743008370688, 0, 1, 24, 759, 27100, 993105, 34617288, 1067280319, 26858490392, 499217336145, 5964692819140, 36882981687519, 79553145323940, 23298085122481, 0
Offset: 0
Examples
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) ); ...
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Crossrefs
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(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),", "));)
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) ).
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