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EXAMPLE
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G.f.: A(x) = 1 + x + 7*x^2 + 105*x^3 + 2366*x^4 + 68776*x^5 + 2390230*x^6 + 95166058*x^7 + 4228436480*x^8 + 206090296497*x^9 + ...
RELATED SERIES.
G.f. A(x) = B(x/A(x)) where B(x) = B(x*A(x)) begins:
B(x) = 1 + x + 8*x^2 + 127*x^3 + 2927*x^4 + 85892*x^5 + 2998264*x^6 + 119665415*x^7 + 5325877575*x^8 + ... + b(n)*x^n + ...
such that b(n) = [x^n] (1 + x*A(x)^(2*n+3))^(n+1) / (n+1),
as well as b(n) = [x^n] A(x)^(n+1) / (n+1),
so that b(n) begin:
[1/1, 2/2, 24/3, 508/4, 14635/5, 515352/6, 20987848/7, 957323320/8, ...].
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n+1) begins:
n=0: [1, 1, 7, 105, 2366, 68776, 2390230, 95166058, ...];
n=1: [1, 2, 15, 224, 4991, 143754, 4962161, 196572300, ...];
n=2: [1, 3, 24, 358, 7896, 225396, 7727644, 304572936, ...];
n=3: [1, 4, 34, 508, 11103, 314192, 10699244, 419541832, ...];
n=4: [1, 5, 45, 675, 14635, 410661, 13890275, 541873525, ...];
n=5: [1, 6, 57, 860, 18516, 515352, 17314836, 671984280, ...];
n=6: [1, 7, 70, 1064, 22771, 628845, 20987848, 810313190, ...];
n=7: [1, 8, 84, 1288, 27426, 751752, 24925092, 957323320, ...]; ...
Compare to the table of coefficients in (1 + x*A(x)^(2*n+3))^(n+1):
n=0: [1, 1, 3, 24, 358, 7896, 225396, 7727644, ...];
n=1: [1, 2, 11, 100, 1465, 31070, 859367, 28808972, ...];
n=2: [1, 3, 24, 253, 3780, 77994, 2089024, 68277867, ...];
n=3: [1, 4, 42, 508, 7915, 161316, 4196916, 133476480, ...];
n=4: [1, 5, 65, 890, 14635, 298981, 7602705, 235213110, ...];
n=5: [1, 6, 93, 1424, 24858, 515352, 12914214, 389369448, ...];
n=6: [1, 7, 126, 2135, 39655, 842331, 20987848, 619044602, ...];
n=7: [1, 8, 164, 3048, 60250, 1320480, 32998388, 957323320, ...]; ...
to see that the main diagonals of the tables are the same.
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