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A274740
Table of coefficients in the iterations of Euler's tree function (A000169), as read by antidiagonals.
1
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
OFFSET
0,5
COMMENTS
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.
FORMULA
See formulas at A274390, which is the main entry for this table.
EXAMPLE
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;
...
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 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), ", ")); )
CROSSREFS
Cf. A274390.
Sequence in context: A338858 A201897 A246658 * A360657 A327350 A137452
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Jul 04 2016
STATUS
approved