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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.

%I A274741 #13 Jul 05 2016 13:03:31
%S A274741 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,
%T A274741 16807,8321,1345,133,11,1,1,1,262144,162463,28396,2729,196,13,1,1,1,
%U A274741 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
%N 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.
%C A274741 See examples and formulas at A274391, which is the main entry for this table.
%C A274741 This entry is the same as table A274391, but read by antidiagonals from top down.
%F A274741 See formulas at A274391, which is the main entry for this table.
%e A274741 See examples at A274391, which is the main entry for this table.
%e A274741 This table begins:
%e A274741 1, 1,  1,   1,     1,       1,         1,          1, ...;
%e A274741 1, 1,  3,  16,   125,    1296,     16807,     262144, ...;
%e A274741 1, 1,  5,  43,   525,    8321,    162463,    3774513, ...;
%e A274741 1, 1,  7,  82,  1345,   28396,    734149,   22485898, ...;
%e A274741 1, 1,  9, 133,  2729,   71721,   2300485,   87194689, ...;
%e A274741 1, 1, 11, 196,  4821,  151376,   5787931,  261066156, ...;
%e A274741 1, 1, 13, 271,  7765,  283321,  12567187,  656778529, ...;
%e A274741 1, 1, 15, 358, 11705,  486396,  24539593, 1457297878, ...;
%e A274741 ...
%e A274741 This table may also be written as a triangle:
%e A274741 1;
%e A274741 1, 1;
%e A274741 1, 1, 1;
%e A274741 1, 3, 1, 1;
%e A274741 1, 16, 5, 1, 1;
%e A274741 1, 125, 43, 7, 1, 1;
%e A274741 1, 1296, 525, 82, 9, 1, 1;
%e A274741 1, 16807, 8321, 1345, 133, 11, 1, 1;
%e A274741 1, 262144, 162463, 28396, 2729, 196, 13, 1, 1;
%e A274741 1, 4782969, 3774513, 734149, 71721, 4821, 271, 15, 1, 1;
%e A274741 1, 100000000, 101808185, 22485898, 2300485, 151376, 7765, 358, 17, 1, 1;
%e A274741 ...
%o A274741 (PARI) {ITERATE(F, n, k) = my(G=x +x*O(x^k)); for(i=1, n, G=subst(G, x, F)); G}
%o A274741 {T(n, k) = my(TREE = serreverse(x*exp(-x +x*O(x^k)))); k!*polcoeff(exp(ITERATE(TREE, n, k)), k)}
%o A274741 /* Print this table as a rectangular array */
%o A274741 for(n=0, 10, for(k=0, 10, print1(T(n, k), ", ")); print(""))
%o A274741 /* Print this table as a triangle */
%o A274741 for(n=0, 12, for(k=0, n, print1(T(k, n-k), ", "));print("") )
%o A274741 /* Print this table as a flattened array */
%o A274741 for(n=0, 12, for(k=0, n, print1(T(k, n-k), ", ")); )
%Y A274741 Cf. A274391.
%K A274741 nonn,tabl
%O A274741 0,8
%A A274741 _Paul D. Hanna_, Jul 04 2016