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
Keywords
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 1..230
Programs
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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)}
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