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%I #9 Jul 05 2016 13:01:49
%S 1,0,1,0,2,1,0,9,4,1,0,64,30,6,1,0,625,332,63,8,1,0,7776,4880,948,108,
%T 10,1,0,117649,89742,18645,2056,165,12,1,0,2097152,1986124,454158,
%U 50680,3800,234,14,1,0,43046721,51471800,13221075,1537524,112625,6324,315,16,1,0,1000000000,1530489744,448434136,55494712,4090980,219000,9772,408,18,1,0,25937424601,51395228090,17386204761,2325685632,176238685,9266706,387205,14288,513,20,1
%N Table of coefficients in the iterations of Euler's tree function (A000169), as read by antidiagonals.
%C See examples and formulas at A274390, which is the main entry for this table.
%C This entry is the same as table A274390, but read by antidiagonals from top down.
%F See formulas at A274390, which is the main entry for this table.
%e See examples at A274390, which is the main entry for this table.
%e This table begins:
%e 1, 0, 0, 0, 0, 0, 0, ...;
%e 1, 2, 9, 64, 625, 7776, 117649, ...;
%e 1, 4, 30, 332, 4880, 89742, 1986124, ...;
%e 1, 6, 63, 948, 18645, 454158, 13221075, ...;
%e 1, 8, 108, 2056, 50680, 1537524, 55494712, ...;
%e 1, 10, 165, 3800, 112625, 4090980, 176238685, ...;
%e 1, 12, 234, 6324, 219000, 9266706, 463975764, ...;
%e 1, 14, 315, 9772, 387205, 18704322, 1067280319, ...;
%e 1, 16, 408, 14288, 637520, 34617288, 2217367600, ...;
%e ...
%e This table may also be written as a triangle:
%e 1;
%e 0, 1;
%e 0, 2, 1;
%e 0, 9, 4, 1;
%e 0, 64, 30, 6, 1;
%e 0, 625, 332, 63, 8, 1;
%e 0, 7776, 4880, 948, 108, 10, 1;
%e 0, 117649, 89742, 18645, 2056, 165, 12, 1;
%e 0, 2097152, 1986124, 454158, 50680, 3800, 234, 14, 1;
%e 0, 43046721, 51471800, 13221075, 1537524, 112625, 6324, 315, 16, 1;
%e 0, 1000000000, 1530489744, 448434136, 55494712, 4090980, 219000, 9772, 408, 18, 1, 0;
%e ...
%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(ITERATE(TREE, n, k), k)}
%o /* Print this table as a rectangular array */
%o for(n=0, 10, for(k=1, 10, print1(T(n, k), ", ")); print(""))
%o /* Print this table as a triangle */
%o for(n=1, 12, for(k=0, n-1, print1(T(k, n-k), ", "));print("") )
%o /* Print this table as a flattened array */
%o for(n=0, 12, for(k=0, n-1, print1(T(k, n-k), ", ")); )
%Y Cf. A274390.
%K nonn,tabl
%O 0,5
%A _Paul D. Hanna_, Jul 04 2016
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