cp's OEIS Frontend

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

%I A141754 #9 Mar 17 2024 07:40:27
%S A141754 1,2,5,16,54,190,684,2505,9291,34793,131277,498312,1900862,7280732,
%T A141754 27983141,107868939,416869874,1614609446,6265869591,24358231651,
%U A141754 94837442844,369755864507,1443420649734,5641088964709,22068939198169
%N A141754 Secondary diagonal of A139755, the table of q-derangement numbers of type A.
%H A141754 Vaclav Kotesovec, <a href="/A141754/b141754.txt">Table of n, a(n) for n = 1..230</a>
%F A141754 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.
%F A141754 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
%o A141754 (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)}
%Y A141754 Cf. A139755, A141753.
%K A141754 nonn
%O A141754 1,2
%A A141754 _Paul D. Hanna_, Jul 05 2008