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EXAMPLE
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The function W that satisfies: W(x,q) = exp( q*x*W(q*x,q) ) begins:
W(x,q) = 1 + q*x + (1/2 + q)*q^2*x^2 +
(1/6 + 1*q + 1/2*q^2 + 1*q^3)*q^3*x^3 +
(1/24 + 1/2*q + 1*q^2 + 7/6*q^3 + 1*q^4 + 1/2*q^5 + 1*q^6)*q^4*x^4 +...
Coefficients of q in {[x^n] W(x,q)} tend to a limit when read backwards:
n=1: (1/2 + q)*q^2 read backwards: [1, 1/2];
n=2: (1/6 + 1*q + 1/2*q^2 + 1*q^3)*q^3 read backwards: [1, 1/2, 1, 1/6];
n=3: (1/24 + 1/2*q + 1*q^2 + 7/6*q^3 + 1*q^4 + 1/2*q^5 + 1*q^6)*q^4 read backwards: [1, 1/2, 1, 7/6, 1, 1/2, 1/24].
The limit of fractional coefficients may be formed into a triangle:
1,
1/2, 1,
7/6, 2, 2,
85/24, 11/3, 65/12, 19/3,
357/40, 19/2, 111/8, 123/8, 81/4, 16891/720, ...
When row n=1,2,3,.. is multiplied by n! we obtain this integer triangle:
1;
1, 2;
7, 12, 12;
85, 88, 130, 152;
1071, 1140, 1665, 1845, 2430;
16891, 21786, 24501, 32066, 36066, 45222;
363378, 450506, 509110, 631883, 718914, 866306, 991571;
9545369, 10821336, 13004356, 14732096, 17438450, 19851112, 23380260, 26447976;
279725995, 316750608, 368695521, 417632601, 484621893, 546334029, 632562585, 713249235, 820357488;
9251279911, 10612100290, 11923578775, 13648746400, 15329052835, 17462968972, 19598497945, 22282099420, 24949824310, 28305482450; ...
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