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A185895
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Exponential generating function is (1-x^1/1!)(1-x^2/2!)(1-x^3/3!)....
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7
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1, -1, -1, 2, 3, 14, -40, -43, -357, -1762, 8004, 13067, 78540, 492439, 3932305, -26867293, -44643557, -363632466, -1729625764, -15939972937, -145669871232, 1488599170613, 3515325612655, 26765194180353, 151925998229148
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OFFSET
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0,4
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COMMENTS
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Conjectures: 1) a(n) differs in sign from a(n-1) iff n is a triangular number (checked up to n = 1225 = (50*51)/2)
2) The same property holds for the coefficients of A(x)^2, the square of the o.g.f. A(x) = 1 - x - x^2 + 2*x^3 + 3*x^4 + ... : A(x)^2 = 1 - 2*x - x^2 + 6*x^3 + 3*x^4 + 18*x^5 - 110*x^6 - 22*x^7 - 483*x^8 - 2800*x^9 + 20030*x^10 + ....
3) The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k. (End)
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LINKS
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FORMULA
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E.g.f.: Product_{k>0} (1 - x^k/k!).
a(n) = Sum_{k=1..n} (n-1)!/(n-k)!*b(k)*a(n-k), where b(k) = Sum_{d divides k} -d*d!^(-k/d) and a(0) = 1 [cf. Vladeta Jovovic's formula in A007837].
E.g.f.: exp(-Sum_{k>=1} Sum_{j>=1} x^(j*k)/(k*(j!)^k)). - Ilya Gutkovskiy, Jun 18 2018
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PROG
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(PARI) {a(n) = if( n<0, 0, n! * polcoeff( prod( k=1, n, 1 - x^k / k!, 1 + x * O(x^n)), n))}
(PARI) {a(n)=if(n<0, 0, if(n==0, 1, sum(k=1, n, (n-1)!/(n-k)!*a(n-k)*sumdiv(k, d, -d*d!^(-k/d)))))} [Hanna]
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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