|
%I #23 Mar 18 2022 13:07:12
%S 1,-1,-1,2,3,14,-40,-43,-357,-1762,8004,13067,78540,492439,3932305,
%T -26867293,-44643557,-363632466,-1729625764,-15939972937,
%U -145669871232,1488599170613,3515325612655,26765194180353,151925998229148
%N Exponential generating function is (1-x^1/1!)(1-x^2/2!)(1-x^3/3!)....
%C From _Peter Bala_, Mar 17 2022: (Start)
%C 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)
%C 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 + ....
%C 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)
%H Seiichi Manyama, <a href="/A185895/b185895.txt">Table of n, a(n) for n = 0..300</a>
%F E.g.f.: Product_{k>0} (1 - x^k/k!).
%F 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].
%F E.g.f.: exp(-Sum_{k>=1} Sum_{j>=1} x^(j*k)/(k*(j!)^k)). - _Ilya Gutkovskiy_, Jun 18 2018
%o (PARI) {a(n) = if( n<0, 0, n! * polcoeff( prod( k=1, n, 1 - x^k / k!, 1 + x * O(x^n)), n))}
%o (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]
%Y Cf. A005651, A007837, A168268.
%K sign
%O 0,4
%A _Michael Somos_, Feb 05 2011
|
