OFFSET
0,4
COMMENTS
Here Laplace( Sum_{n>=0} c(n)*x^n/n! ) = Sum_{n>=0} c(n)*x^n.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..220 (terms 0..100 from Paul D. Hanna)
EXAMPLE
E.g.f.: A(x) = 1 + x + x^2/2! + 2*x^3/3! + 4*x^4/4! + 12*x^5/5! + 38*x^6/6! +...
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, 1, 2, 4, 12, 38, 150, 648, ...];
n=2: [1, 2, 4, 10, 30, 104, 420, 1896, 9632, ...];
n=3: [1, 3, 9, 30, 114, 486, 2316, 12210, 70632, ...];
n=4: [1, 4, 16, 68, 316, 1608, 8936, 54024, 353496, ...];
n=5: [1, 5, 25, 130, 720, 4280, 27330, 187230, 1372640, ...];
n=6: [1, 6, 36, 222, 1434, 9792, 70908, 544800, 4437288, ...];
n=7: [1, 7, 49, 350, 2590, 20034, 162680, 1389066, 12474672, ...];
n=8: [1, 8, 64, 520, 4344, 37616, 339216, 3193200, 31407632, ...];
n=9: [1, 9, 81, 738, 6876, 65988, 655326, 6752934, 72308376, ...]; ...
such that the alternating sums of antidiagonals vanish for n>1:
1 - 1 = 0;
1 - 2 + 1 = 0;
2 - 4 + 3 - 1 = 0;
4 - 10 + 9 - 4 + 1 = 0;
12 - 30 + 30 - 16 + 5 - 1 = 0;
38 - 104 + 114 - 68 + 25 - 6 + 1 = 0; ...
PROG
(PARI) {a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0); A[#A]=Vec(sum(m=1, #A, serlaplace(serconvol(Ser(A), exp(x+O(x^#A)))^m)*(-x)^m))[#A]); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 06 2012
STATUS
approved