| To illustrate the recurrence, list coefficients of A(x^2)^(2n+1):
A^1: . 1,... 1,... 2,... 10,... 30,... 131,.......;
A^3: .... 1,... 3,... 9,... 43,... 168,... 735, ...;
A^5: ....... 1,... 5,... 20,... 100,... 455,.......;
A^7: .......... 1,... 7,... 35,... 189,... 959, ...;
A^9: ............. 1,... 9,... 54,... 318,.......;
A^11: ............... 1,... 11,... 77,... 495, ...;
A^13: .................. 1,... 13,... 104,.......;
A^15: ..................... 1,... 15,... 135, ...;
A^17: ........................ 1,... 17,.......;
A^19: ........................... 1,... 19, ...;
A^21: .............................. 1,.......;
A^23: ................................. 1, ...;...
then sum the squares of the coefficients in each column:
a(0) = 1^2 = 1;
a(1) = 1^2 = 1;
a(2) = 1^2 + 1^2 = 2;
a(3) = 3^2 + 1^2 = 10;
a(4) = 2^2 + 5^2 + 1^2 = 30;
a(5) = 9^2 + 7^2 + 1^2 = 131;
a(6) = 10^2 + 20^2 + 8^2 + 1^2 = 582;
a(7) = 43^2 + 35^2 + 11^2 + 1^2 = 3196;
a(8) = 30^2 + 100^2 + 54^2 + 13^2 + 1^2 = 13986;
a(9) = 168^2 + 189^2 + 77^2 + 15^2 + 1^2 = 70100.
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