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EXAMPLE
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G.f.: A(x) = 1 + x + 2*x^2 + 11*x^3 + 140*x^4 + 2898*x^5 + 80844*x^6 + 2786091*x^7 + 113184008*x^8 + 5266198778*x^9 + 275248731860*x^10 + ...
ILLUSTRATION OF THE DEFINITION.
The table of coefficients of x^k in A(x)^(n^2) begins:
n=1: [1, 1, 2, 11, 140, 2898, 80844, ...];
n=2: [1, 4, 14, 72, 741, 13724, 364546, ...];
n=3: [1, 9, 54, 327, 2826, 42660, 1017720, ...];
n=4: [1, 16, 152, 1216, 10540, 129376, 2559792, ...];
n=5: [1, 25, 350, 3775, 37750, 427480, 6820800, ...];
n=6: [1, 36, 702, 10056, 123165, 1477980, 20712546, ...];
n=7: [1, 49, 1274, 23667, 359856, 4953998, 69355972, ...]; ...
in which the main diagonal
D0 = [1, 4, 54, 1216, 37750, 1477980, 69355972, 3775816704, ...]
and the adjacent diagonal
D1 = [1, 9, 152, 3775, 123165, 4953998, 235988544, 12954335103, ...]
are related by D0[n-1] = 2*n*D1[n-2] for n>=2.
The related sequence D0[n-1]/n^2, n>=1, begins:
[1, 1, 6, 76, 1510, 41055, 1415428, 58997136, 2878741134, 160698224230, ...].
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