A141754 Secondary diagonal of A139755, the table of q-derangement numbers of type A.

1, 2, 5, 16, 54, 190, 684, 2505, 9291, 34793, 131277, 498312, 1900862, 7280732, 27983141, 107868939, 416869874, 1614609446, 6265869591, 24358231651, 94837442844, 369755864507, 1443420649734, 5641088964709, 22068939198169

Offset

1,2

Links

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

Formula

a(n) = [q^n] { ([n+2]_q)! * Sum_{m=0..n+2} (-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), with ([0]_q)! = 1.

a(n) ~ c * 4^n / sqrt(Pi*n), where c = 2*QPochhammer(1/2)^2 = 0.166797127727497043069083616310625... - Vaclav Kotesovec, Aug 30 2023, updated Mar 17 2024

Prog

(PARI) {a(n)=polcoeff(prod(j=1, n+2, (1-q^j)/(1-q))* sum(k=0, n+2, (-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, A141753.

Sequence in context: A120899 A149960 A149961 * A149962 A149963 A149964

Adjacent sequences: A141751 A141752 A141753 * A141755 A141756 A141757

Keyword

nonn

Author

Paul D. Hanna, Jul 05 2008

Status

approved