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A129481
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a(n) = coefficient of x^n in n!*Product_{k=0..n} [Sum_{j=0..k} x^j/j! ].
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2
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1, 1, 3, 19, 175, 2111, 31321, 550810, 11194177, 258068893, 6653230111, 189653427206, 5922604033567, 201075967613262, 7373834652641003, 290474615891145106, 12232735359488840833, 548429151685677131389
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OFFSET
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0,3
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COMMENTS
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LINKS
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FORMULA
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a(n) ~ c * n^n, where c = 0.660942456683588459181273625114230472913... . - Vaclav Kotesovec, Feb 10 2015
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EXAMPLE
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a(2) = [x^2] 2!*(1)*(1+x)*(1+x+x^2/2!) = [x^2] (2 +4*x +3*x^2 +x^3) = 3.
a(3) = [x^3] 3!*(1)*(1+x)*(1 + x + x^2/2!)*(1 + x + x^2/2! + x^3/3!) =
[x^3] (6 + 18*x + 24*x^2 + 19*x^3 +...) = 19.
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MATHEMATICA
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Flatten[{1, Table[Coefficient[Expand[n!*Product[Sum[x^j/j!, {j, 0, k}], {k, 0, n}]], x^n], {n, 1, 20}]}] (* Vaclav Kotesovec, Feb 10 2015 *)
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PROG
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(PARI) {a(n)=n!*polcoeff(prod(k=0, n, sum(j=0, k, x^j/j!)+x*O(x^n)), n)}
(Magma)
m:=30; R<x>:=PowerSeriesRing(Integers(), m+2);
p:= func< n, x | (&*[ (&+[x^j/Factorial(j): j in [0..k]]) : k in [0..n]]) >;
A129481:= func< n | Coefficient(R!(Laplace( p(n, x) )), n) >;
(SageMath)
def p(n, x): return product(sum(x^j/factorial(j) for j in range(k+1)) for k in range(n+1))
def A129481(n): return factorial(n)*( p(n, x) ).series(x, 101).list()[n]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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