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A209903
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E.g.f.: Product_{n>=1} B(x^n) where B(x) = exp(exp(x)-1) = e.g.f. of Bell numbers.
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11
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1, 1, 4, 17, 111, 752, 6893, 64171, 733540, 8751579, 119847295, 1716294780, 27583937857, 460405876777, 8428298492136, 160944930254405, 3309210789416387, 70814345769448444, 1617322515279759301, 38322855872232745163, 960820910852189283072
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OFFSET
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0,3
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LINKS
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FORMULA
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E.g.f.: exp( Sum_{n>=1} x^n/n! / (1-x^n) ).
E.g.f.: exp( Sum_{n>=1} A057625(n)*x^n/n! ).
E.g.f.: exp( Sum_{n>=1} exp(x^n)-1 ).
a(n) = (n-1)! * Sum_{k=1..n} k * (Sum_{d|k} 1/d!) * a(n-k)/(n-k)! for n > 0. - Seiichi Manyama, Jul 02 2021
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EXAMPLE
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E.g.f.: A(x) = 1 + x + 4*x^2/2! + 17*x^3/3! + 111*x^4/4! + 752*x^5/5! +...
Let B(x) = exp(exp(x)-1) be the e.g.f. of Bell numbers:
B(x) = 1 + x + 2*x^2/2! + 5*x^3/3! + 15*x^4/4! + 52*x^5/5! + 203*x^6/6! +...
then the e.g.f. of this sequence equals the infinite product:
A(x) = B(x)*B(x^2)*B(x^3)*B(x^4)*B(x^5)*B(x^6)...
The logarithm of the e.g.f. A(x) begins:
log(A(x)) = x + 3*x^2/2! + 7*x^3/3! + 37*x^4/4! + 121*x^5/5! + 1201*x^6/6! +...+ A057625(n)*x^n/n! +...
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PROG
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(PARI) {a(n)=local(Bell=exp(exp(x+x*O(x^n))-1)); n!*polcoeff(prod(m=1, n, subst(Bell, x, x^m+x*O(x^n))), n)}
(PARI) {a(n)=n!*polcoeff(exp(sum(m=1, n, x^m/m!/(1-x^m+x*O(x^n))), n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) a(n) = if(n==0, 1, (n-1)!*sum(k=1, n, k*sumdiv(k, d, 1/d!)*a(n-k)/(n-k)!)); \\ Seiichi Manyama, Jul 02 2021
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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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