A139755 -id:A139755 - cp's OEIS frontend

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

Paul D. Hanna, Jul 05 2008

nonn

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

Cf. A139755, A141754.

{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)}

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

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

Original entry on oeis.org

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

Paul D. Hanna, Jul 05 2008

nonn

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

Cf. A139755, A141753.

{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)}

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

A140075 Table of q-derangement numbers of type B, by rows.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 3, 4, 5, 5, 4, 4, 2, 1, 1, 4, 8, 13, 18, 22, 26, 28, 28, 25, 21, 17, 11, 7, 3, 1

Offset: 1

Jonathan Vos Post, Jun 14 2008

This sequence is from Table 1.2 of Chen and Wang, p. 3.

			The table begins:
===================================================================================
k=...|.1.|.2.|.3.|..4.|..5.|..6.|..7.|..8.|..9.|.10.|.11.|.12.|.13.|.14.|.15.|.16.|
===================================================================================
n=1..|.1.|
n=2..|.1.|.2.|.1.|..1.|
n=3..|.1.|.3.|.4.|..5.|..5.|..4.|..4.|..2.|..1.|
n=4..|.1.|.4.|.8.|.13.|.18.|.22.|.26.|.28.|.28.|.25.|.21.|.17.|.11.|..7.|..3.|..1.|
===================================================================================

William Y. C. Chen and David G. L. Wang, The Limiting Distributions of the Coefficients of the q-Derangement Numbers

Cf. A000166, A139755.

cp's OEIS Frontend

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

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

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A140075 Table of q-derangement numbers of type B, by rows.

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