A141753 Main diagonal of A139755, the table of q-derangement numbers of type A.

1, 1, 2, 7, 24, 86, 313, 1157, 4325, 16303, 61856, 235917, 903620, 3473381, 13391280, 51761781, 200523644, 778342906, 3026400508, 11785538461, 45959004812, 179444813270, 701422450293, 2744562302533, 10749124666643, 42135320616531

Offset: 1

Author

Paul D. Hanna, Jul 05 2008

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..230

Crossrefs

Cf. A139755, A141754.

Programs

PARI

{a(n)=polcoeff(prod(j=1,n+1,(1-q^j)/(1-q))* sum(k=0,n+1,(-1)^k*q^(k*(k-1)/2)/if(k==0,1,prod(j=1,k,(1-q^j)/(1-q)))),n,q)}

Formula

a(n) = [q^n] { ([n+1]q)! * Sum{m=0..n+1} (-1)^m * q^(m(m-1)/2) / ([m]q)! }; here, the q-factorial of n is denoted by ([n]_q)! = Product{j=1..n} (1-q^j)/(1-q).

a(n) ~ c * 4^n / sqrt(Pi*n), where c = A048651^2 = QPochhammer(1/2)^2 = 0.083398563863748521534541808155312... - Vaclav Kotesovec, Aug 30 2023