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%I #4 Mar 30 2012 18:36:50
%S 1,1,1,1,2,1,1,1,3,1,1,4,3,2,1,1,2,1,2,5,1,1,6,5,4,3,2,1,1,1,3,3,5,3,
%T 7,1,1,8,7,2,5,4,3,2,1,1,2,3,4,1,3,7,4,9,1,1,10,9,8,7,6,5,4,3,2,1,1,1,
%U 3,1,5,6,7,2,3,5,11,1,1,12,11,10,9,8,7,6,5,4,3,2,1,1,1,3,4,5,3,1,4,9,10,11
%N Consider the array T(n, m) where the n-th row is the sequence of integer coefficients of A(x), where 1<=a(n)<=n, such that A(x)^(1/n) consists entirely of integer coefficients and where m is the (m+1)-th coefficient. This is the antidiagonal read zig-zag.
%C T(n,n)=T(n,n+2)=A111627.
%e Table begins
%e \k...0...1....2....3....4....5....6....7....8....9...10...11...12...13
%e n\
%e 1| 1 1 1 1 1 1 1 1 1 1 1 1 1 1
%e 2| 1 2 1 2 2 2 1 2 2 2 1 2 1 2
%e 3| 1 3 3 1 3 3 3 3 3 3 3 3 1 3
%e 4| 1 4 2 4 3 4 4 4 1 4 4 4 3 4
%e 5| 1 5 5 5 5 1 5 5 5 5 4 5 5 5
%e 6| 1 6 3 2 3 6 6 6 3 4 6 6 6 6
%e 7| 1 7 7 7 7 7 7 1 7 7 7 7 7 7
%e 8| 1 8 4 8 2 8 4 8 7 8 8 8 4 8
%e 9| 1 9 9 3 9 9 3 9 9 1 9 9 6 9
%e 10| 1 10 5 10 10 2 5 10 10 10 3 10 5 10
%e 11| 1 11 11 11 11 11 11 11 11 11 11 1 11 11
%e 12| 1 12 6 4 9 12 4 12 12 8 6 12 6 12
%e 13| 1 13 13 13 13 13 13 13 13 13 13 13 13 1
%e 14| 1 14 7 14 7 14 14 2 7 14 14 14 14 14
%e 15| 1 15 15 5 15 3 10 15 15 10 15 15 5 15
%e 16| 1 16 8 16 4 16 8 16 10 16 8 16 12 16
%t f[n_] := f[n] = Block[{a}, a[0] = 1; a[l_] := a[l] = Block[{k = 1, s = Sum[ a[i]*x^i, {i, 0, l - 1}]}, While[ IntegerQ[ Last[ CoefficientList[ Series[(s + k*x^l)^(1/n), {x, 0, l}], x]]] != True, k++ ]; k]; Table[a[j], {j, 0, 32}]]; g[n_, m_] := f[n][[m]];
%Y Cf. A111613, A083952, A083953, A083954, A083945, A083946, A083947, A083948, A083949, A083950, A084066, A084067.
%Y Cf. A109626, A111603.
%K nonn,tabl
%O 1,5
%A _Paul D. Hanna_ and _Robert G. Wilson v_, Aug 01 2005
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