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EXAMPLE
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E.g.f.: A(x) = 1 + x + 2*x^2/2! + 5*x^3/3! + 15*x^4/4! + 54*x^5/5! +...
The table of coefficients in the Laplace transform of A(x)^n begins:
n=0: [1, 0, 0, 0, 0, 0, 0, 0, 0, ...];
n=1: [1, 1, 2, 5, 15, 54, 231, 1151, 6544, ...];
n=2: [1, 2, 6, 22, 94, 458, 2510, 15322, 103366, ...];
n=3: [1, 3, 12, 57, 309, 1872, 12507, 91317, 723738, ...];
n=4: [1, 4, 20, 116, 756, 5436, 42612, 361100, 3287428, ...];
n=5: [1, 5, 30, 205, 1555, 12890, 115535, 1111195, 11401900, ...];
n=6: [1, 6, 42, 330, 2850, 26694, 268626, 2884326, 32869914, ...];
n=7: [1, 7, 56, 497, 4809, 50148, 558915, 6616337, 82785766, ...];
n=8: [1, 8, 72, 712, 7624, 87512, 1068872, 13813432, 188042728, ...];
n=9: [1, 9, 90, 981,11511,144126, 1912887, 26774775, 393614568, ...]; ...
where the antidiagonal sums equal this sequence:
1 + 1 + 0 = 2;
1 + 2 + 2 + 0 = 5;
1 + 3 + 6 + 5 + 0 = 15;
1 + 4 + 12 + 22 + 15 + 0 = 54; ...
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PROG
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(PARI) a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, serlaplace(serconvol(A+x*O(x^n), exp(x+x*O(x^n)))^m+x*O(x^n))*x^m)+x*O(x^n)); polcoeff(A, n)
for(n=0, 20, print1(a(n), ", "))
(PARI) /* From e.g.f. involving iterated integration: */
{INTEGRATE(n, F)=local(G=F); for(i=1, n, G=intformal(G)); G}
{a(n)=local(A=1+x); for(i=1, n, A=1+sum(k=1, n, INTEGRATE(k, (A+x*O(x^n))^k))); n!*polcoeff(A, n)}
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