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

1,3

Links

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

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

Prog

(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)}

Crossrefs

Cf. A139755, A141754.

Sequence in context: A297345 A052986 A053368 * A014300 A128086 A131824

Adjacent sequences: A141750 A141751 A141752 * A141754 A141755 A141756

Keyword

nonn

Author

Paul D. Hanna, Jul 05 2008

Status

approved