A185895 Exponential generating function is (1-x^1/1!)(1-x^2/2!)(1-x^3/3!)....

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

Offset: 0

Author

Michael Somos, Feb 05 2011

Keywords

• sign

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

• Seiichi Manyama, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A005651, A007837, A168268.

Programs

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]

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