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

0,8

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.

Links

Table of n, a(n) for n=0..90.

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

Adjacent sequences: A274738 A274739 A274740 * A274742 A274743 A274744

Keyword

nonn,tabl

Author

Paul D. Hanna, Jul 04 2016

Status

approved