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A274740 Table of coefficients in the iterations of Euler's tree function (A000169), as read by antidiagonals.

%I A274740 #9 Jul 05 2016 13:01:49
%S A274740 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 A274740 10,1,0,117649,89742,18645,2056,165,12,1,0,2097152,1986124,454158,
%U A274740 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 A274740 Table of coefficients in the iterations of Euler's tree function (A000169), as read by antidiagonals.
%C A274740 See examples and formulas at A274390, which is the main entry for this table.
%C A274740 This entry is the same as table A274390, but read by antidiagonals from top down.
%F A274740 See formulas at A274390, which is the main entry for this table.
%e A274740 See examples at A274390, which is the main entry for this table.
%e A274740 This table begins:
%e A274740 1,  0,   0,     0,       0,        0,          0, ...;
%e A274740 1,  2,   9,    64,     625,     7776,     117649, ...;
%e A274740 1,  4,  30,   332,    4880,    89742,    1986124, ...;
%e A274740 1,  6,  63,   948,   18645,   454158,   13221075, ...;
%e A274740 1,  8, 108,  2056,   50680,  1537524,   55494712, ...;
%e A274740 1, 10, 165,  3800,  112625,  4090980,  176238685, ...;
%e A274740 1, 12, 234,  6324,  219000,  9266706,  463975764, ...;
%e A274740 1, 14, 315,  9772,  387205, 18704322, 1067280319, ...;
%e A274740 1, 16, 408, 14288,  637520, 34617288, 2217367600, ...;
%e A274740 ...
%e A274740 This table may also be written as a triangle:
%e A274740 1;
%e A274740 0, 1;
%e A274740 0, 2, 1;
%e A274740 0, 9, 4, 1;
%e A274740 0, 64, 30, 6, 1;
%e A274740 0, 625, 332, 63, 8, 1;
%e A274740 0, 7776, 4880, 948, 108, 10, 1;
%e A274740 0, 117649, 89742, 18645, 2056, 165, 12, 1;
%e A274740 0, 2097152, 1986124, 454158, 50680, 3800, 234, 14, 1;
%e A274740 0, 43046721, 51471800, 13221075, 1537524, 112625, 6324, 315, 16, 1;
%e A274740 0, 1000000000, 1530489744, 448434136, 55494712, 4090980, 219000, 9772, 408, 18, 1, 0;
%e A274740 ...
%o A274740 (PARI) {ITERATE(F, n, k) = my(G=x +x*O(x^k)); for(i=1, n, G=subst(G, x, F)); G}
%o A274740 {T(n, k) = my(TREE = serreverse(x*exp(-x +x*O(x^k)))); k!*polcoeff(ITERATE(TREE, n, k), k)}
%o A274740 /* Print this table as a rectangular array */
%o A274740 for(n=0, 10, for(k=1, 10, print1(T(n, k), ", ")); print(""))
%o A274740 /* Print this table as a triangle */
%o A274740 for(n=1, 12, for(k=0, n-1, print1(T(k, n-k), ", "));print("") )
%o A274740 /* Print this table as a flattened array */
%o A274740 for(n=0, 12, for(k=0, n-1, print1(T(k, n-k), ", ")); )
%Y A274740 Cf. A274390.
%K A274740 nonn,tabl
%O A274740 0,5
%A A274740 _Paul D. Hanna_, Jul 04 2016