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