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EXAMPLE
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G.f.: A(x) = 1 + 2*x + 6*x^2 + 32*x^3 + 240*x^4 + 2232*x^5 + 24080*x^6 +...
The table of coefficients in A(x)^n begins:
n=1: [1, 2, 6, 32, 240, 2232, 24080, 290048, 3809088, ...];
n=2: [1, 4, 16, 88, 644, 5808, 60992, 718560, 9267776, ...];
n=3: [1, 6, 30, 176, 1284, 11328, 116136, 1339200, 16962240, ...];
n=4: [1, 8, 48, 304, 2248, 19584, 196800, 2224256, 27672720, ...];
n=5: [1, 10, 70, 480, 3640, 31592, 312640, 3470080, 42432080, ...];
n=6: [1, 12, 96, 712, 5580, 48624, 476224, 5203680, 62599152, ...];
n=7: [1, 14, 126, 1008, 8204, 72240, 703640, 7590592, 89949552, ...]; ...
where a(n) is obtained from the antidiagonals in the above table like so:
a(1) = 2*(1*1/1);
a(2) = 2*(1*2/1 + 2*1/2);
a(3) = 2*(1*6/1 + 4*4/2 + 6*1/3);
a(4) = 2*(1*32/1 + 6*16/2 + 16*6/3 + 32*1/4);
a(5) = 2*(1*240/1 + 8*88/2 + 30*30/3 + 88*8/4 + 240*1/5);
a(6) = 2*(1*2232/1 + 10*644/2 + 48*176/3 + 176*48/4 + 644*10/5 + 2232*1/6);
a(7) = 2*(1*24080/1 + 12*5808/2 + 70*1284/3 + 304*304/4 + 1284*70/5 + 5808*12/6 + 24080*1/7); ...
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PROG
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(PARI) a(n)=local(A=1 + sum(j=1, n-1, a(j)*x^j)+x*O(x^n)); if(n==0, 1, 2*sum(k=0, n-1, polcoeff(A^(n-k), k)*polcoeff(A^(k+1)/(k+1), n-k-1)))
for(n=0, 25, print1(a(n), ", "))
(PARI) a(n)=local(A=1+x); for(i=1, n, A=1+2*subst(intformal(A), x, serreverse(x/A +x*O(x^n)))); polcoeff(A, n)
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