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A274741
Table of coefficients in functions that satisfy W_n(x) = W_{n-1}(x)^W_n(x), with W_0(x) = exp(x), as read by antidiagonals.
1
1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 16, 5, 1, 1, 1, 125, 43, 7, 1, 1, 1, 1296, 525, 82, 9, 1, 1, 1, 16807, 8321, 1345, 133, 11, 1, 1, 1, 262144, 162463, 28396, 2729, 196, 13, 1, 1, 1, 4782969, 3774513, 734149, 71721, 4821, 271, 15, 1, 1, 1, 100000000, 101808185, 22485898, 2300485, 151376, 7765, 358, 17, 1, 1, 1, 2357947691, 3129525793, 796769201, 87194689, 5787931, 283321, 11705, 457, 19, 1, 1, 1, 61917364224, 108063152091, 32084546824, 3815719969, 261066156, 12567187, 486396, 16785, 568, 21, 1, 1
OFFSET
0,8
COMMENTS
See examples and formulas at A274391, which is the main entry for this table.
This entry is the same as table A274391, but read by antidiagonals from top down.
FORMULA
See formulas at A274391, which is the main entry for this table.
EXAMPLE
See examples at A274391, which is the main entry for this table.
This table begins:
1, 1, 1, 1, 1, 1, 1, 1, ...;
1, 1, 3, 16, 125, 1296, 16807, 262144, ...;
1, 1, 5, 43, 525, 8321, 162463, 3774513, ...;
1, 1, 7, 82, 1345, 28396, 734149, 22485898, ...;
1, 1, 9, 133, 2729, 71721, 2300485, 87194689, ...;
1, 1, 11, 196, 4821, 151376, 5787931, 261066156, ...;
1, 1, 13, 271, 7765, 283321, 12567187, 656778529, ...;
1, 1, 15, 358, 11705, 486396, 24539593, 1457297878, ...;
...
This table may also be written as a triangle:
1;
1, 1;
1, 1, 1;
1, 3, 1, 1;
1, 16, 5, 1, 1;
1, 125, 43, 7, 1, 1;
1, 1296, 525, 82, 9, 1, 1;
1, 16807, 8321, 1345, 133, 11, 1, 1;
1, 262144, 162463, 28396, 2729, 196, 13, 1, 1;
1, 4782969, 3774513, 734149, 71721, 4821, 271, 15, 1, 1;
1, 100000000, 101808185, 22485898, 2300485, 151376, 7765, 358, 17, 1, 1;
...
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(exp(ITERATE(TREE, n, k)), k)}
/* Print this table as a rectangular array */
for(n=0, 10, for(k=0, 10, print1(T(n, k), ", ")); print(""))
/* Print this table as a triangle */
for(n=0, 12, for(k=0, n, print1(T(k, n-k), ", ")); print("") )
/* Print this table as a flattened array */
for(n=0, 12, for(k=0, n, print1(T(k, n-k), ", ")); )
CROSSREFS
Cf. A274391.
Sequence in context: A265025 A284578 A070989 * A135021 A347811 A320412
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Jul 04 2016
STATUS
approved