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A274740
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Table of coefficients in the iterations of Euler's tree function (A000169), as read by antidiagonals.
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1
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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, 10, 1, 0, 117649, 89742, 18645, 2056, 165, 12, 1, 0, 2097152, 1986124, 454158, 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
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OFFSET
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0,5
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COMMENTS
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See examples and formulas at A274390, which is the main entry for this table.
This entry is the same as table A274390, but read by antidiagonals from top down.
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LINKS
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FORMULA
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See formulas at A274390, which is the main entry for this table.
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EXAMPLE
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See examples at A274390, which is the main entry for this table.
This table begins:
1, 0, 0, 0, 0, 0, 0, ...;
1, 2, 9, 64, 625, 7776, 117649, ...;
1, 4, 30, 332, 4880, 89742, 1986124, ...;
1, 6, 63, 948, 18645, 454158, 13221075, ...;
1, 8, 108, 2056, 50680, 1537524, 55494712, ...;
1, 10, 165, 3800, 112625, 4090980, 176238685, ...;
1, 12, 234, 6324, 219000, 9266706, 463975764, ...;
1, 14, 315, 9772, 387205, 18704322, 1067280319, ...;
1, 16, 408, 14288, 637520, 34617288, 2217367600, ...;
...
This table may also be written as a triangle:
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, 10, 1;
0, 117649, 89742, 18645, 2056, 165, 12, 1;
0, 2097152, 1986124, 454158, 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;
...
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PROG
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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 rectangular array */
for(n=0, 10, for(k=1, 10, print1(T(n, k), ", ")); print(""))
/* Print this table as a triangle */
for(n=1, 12, for(k=0, n-1, print1(T(k, n-k), ", ")); print("") )
/* Print this table as a flattened array */
for(n=0, 12, for(k=0, n-1, print1(T(k, n-k), ", ")); )
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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