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EXAMPLE
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Illustration of initial terms.
a(0) = 1;
a(1) = [x^1] (1 + x + x^2/2 + x^3/3 + x^4/4 +...)^1! = 1 ;
a(2) = [x^2] (1 + x + x^2/2 + x^3/3 + x^4/4 +...)^2!, or
a(2) = [x^2] (1 + 2*x + 2*x^2 + 5/3*x^3 + 17/12*x^4 +...) = 2 ;
a(3) = [x^3] (1 + x + x^2/2 + x^3/3 + x^4/4 +...)^3!, or
a(3) = [x^3] (1 + 6*x + 18*x^2 + 37*x^3 + 241/4*x^4 +...) = 37 ;
a(4) = [x^4] (1 + x + x^2/2 + x^3/3 + x^4/4 +...)^4!, or
a(4) = [x^4] (1 + 24*x + 288*x^2 + 2308*x^3 + 13921*x^4 +...) = 13921 ;
...
a(n) = [x^n] (1 + x + x^2/2 + x^3/3 + x^4/4 +...+ x^k/k +...)^n! ;
...
The coefficients of x^k, k=0..n, in (1 - log(1-x))^n! forms the triangle T(n,k):
[1];
[1, 1];
[1, 2, 2];
[1, 6, 18, 37];
[1, 24, 288, 2308, 13921];
[1, 120, 7200, 288020, 8642405, 207504608];
[1, 720, 259200, 62208120, 11197526430, 1612462485648, 193499235977786];
[1, 5040, 12700800, 21337344840, 26885057673810, 27100144537250736, 22764130374754974422, 16390183551007874514674];
[1, 40320, 812851200, 10924720134720, 110121179161192080, 888017192033323164288, 5967475567171901800336816, 34372659584069639646227206672, 173238206541606827885872411575542]; ...
in which the main diagonal forms this sequence: a(n) = T(n,n),
where
T(n,k) = Sum_{j=0..k} binomial(n!, j) * j!/k! * (-1)^(k-j) * Stirling1(k, j).
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PROG
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(PARI) {a(n) = polcoeff( (1 - log(1-x +x*O(x^n)))^n!, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = sum(k=0, n, binomial(n!, k) * k!/n! * (-1)^(n-k) * stirling(n, k, 1) )}
for(n=0, 20, print1(a(n), ", "))
(PARI) {T(n, k) = sum(j=0, k, binomial(n!, j) * j!/k! * (-1)^(k-j) * stirling(k, j, 1) )}
for(n=0, 20, print1(T(n, n), ", "))
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