bibkey: richman2023a053871 authors: Harry Richman year: 2023 title: “OEIS A053871, a(n) = 2*(n-1)*(a(n-1) + a(n-2)), starting a(0) = 1; a(1) = 0; congruence conjecture of Harry Richman” doi: null url: https://oeis.org/A053871 claim: “The FORMULA section of A053871 conjectures that (-1)^m a(m) and (-1)^n a(n) are congruent modulo q whenever q is odd and m and n are congruent modulo q.” strata_touched:

• D5/S3/Combinatorics/DerangedMatchingCongruence license: citation-only triage: anchor

OEIS A053871, Richman’s congruence

A053871 (Cris Moore and Christian G. Bower, 2000) is the sequence

a(n) = 2*(n-1)*(a(n-1) + a(n-2)), starting a(0) = 1; a(1) = 0.

Its COMMENTS read it as the number of deranged matchings of 2n people and as the central moments of the chi-squared distribution:

a(n) is the n-th central moment of a central chi-squared distribution (with 1 degree of freedom), i.e., a(n) = E[ (Y- E[Y] )^n ] = E[ (X^2 - 1 )^n ] where Y is chi-squared, X is std normal, X~N(0,1), and the expectation operator is E[]. - David Fioramonti, May 11 2016

and its FORMULA section states

a(n) = (-1)^nSum_{k=0..n} (-1)^kC(n, k)(2k-1)!!. - Benoit Cloitre, May 01 2003; corrected by David Fioramonti, May 17 2016

Conjecture: if m == n (mod q) for q odd, then (-1)^ma(m) == (-1)^na(n) (mod q). - Harry Richman, Aug 29 2023

The data begin 1, 0, 2, 8, 60, 544, 6040, 79008, 1190672, 20314880. The LINKS include D. B. Costa, B. A. Dobrescu and P. J. Fox, Chiral Abelian gauge theories with few fermions, arXiv:2001.11991 [hep-ph].

Verified locator

• URL: https://oeis.org/A053871 (revision 173, 2026-04-23, FORMULA), retrieved 2026-09-27.