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 A274390 Table of coefficients in the iterations of Euler's tree function (A000169), as read by antidiagonals. 10
 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 (list; table; graph; refs; listen; history; text; internal format)
 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 Cf. A274391, A000169, A207833, A227278; diagonals: A274389, A274392. Cf. A274570 (transforms diagonals). Cf. A274740 (same table, but read differently). Sequence in context: A287318 A173003 A294411 * A244128 A016584 A293961 Adjacent sequences:  A274387 A274388 A274389 * A274391 A274392 A274393 KEYWORD nonn,tabl AUTHOR Paul D. Hanna, Jun 19 2016 STATUS approved

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Last modified June 20 19:36 EDT 2019. Contains 324234 sequences. (Running on oeis4.)