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A185895 Exponential generating function is (1-x^1/1!)(1-x^2/2!)(1-x^3/3!).... 7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
From Peter Bala, Mar 17 2022: (Start)
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)
LINKS
FORMULA
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
PROG
(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]
CROSSREFS
Sequence in context: A203578 A329442 A281486 * A358651 A128849 A294495
KEYWORD
sign
AUTHOR
Michael Somos, Feb 05 2011
STATUS
approved

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Last modified March 28 14:13 EDT 2024. Contains 371254 sequences. (Running on oeis4.)