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, 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

Author

Paul D. Hanna, Jul 04 2016

Keywords

• nonn
• tabl

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.

Examples

			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;
...
		

Crossrefs

Cf. A274391.

Programs

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), ", ")); )

Formula

See formulas at A274391, which is the main entry for this table.