OFFSET
1,2
COMMENTS
a(n) is the number of nonnegative integer matrices with sum of entries equal to 2*n-2 (or 2*n-1), no zero rows or columns, which are centrally symmetric. - Ludovic Schwob, Feb 17 2024
LINKS
Evgeniy Krasko, Counting Unlabelled Chord Diagrams of Maximal Genus, arXiv:1709.00796 [math.CO], 2017.
T. K. Petersen, A two-sided analogue of the Coxeter complex, arXiv:1607.00086 [math.CO], 2016.
EXAMPLE
The a(2) = 5 matrices whose sum of entries is equal to 2:
[2] [1 1]
.
[1] [1 0] [0 1]
[1] [0 1] [1 0]
MAPLE
B:=proc(n) local f;
option remember;
if n=1 then 1+s*t;
elif n>1 then
f:=B(n-1);
RETURN(simplify( (2*n*s*t-s*t+1)*f+(2*s*t*(1-s)+s/n*(1-s)*(1-t))*diff(f, s) + (2*s*t*(1-t)+t/n*(1-s)*(1-t))*diff(f, t) + 2/n*s*t*(1-s)*(1-t)*diff( diff(f, s), t) ));
fi;
end:
seq(eval(eval(subs(s=x/(1+x), t=y/(1+y), B(n))*(1+x)^n*(1+y)^n, y=1), x=1), n=1..30);
MATHEMATICA
B[n_] := B[n] = Which[n == 1, 1 + s*t, n > 1, f = B[n - 1]; Return[ Simplify[ (2*n*s*t - s*t + 1)*f + (2*s*t*(1 - s) + s/n*(1 - s)*(1 - t))*D[f, s] + (2*s*t*(1 - t) + t/n*(1 - s)*(1 - t))*D[f, t] + 2/n*s*t*(1 - s)*(1 - t)*D[ D[f, s], t]]]];
Join[{1}, Table[bn = ((B[n] /. {s -> x/(1 + x), t -> y/(1 + y)})*(1 + x)^n*(1 + y)^n /. {y -> 1, x -> 1}); Print[bn]; bn, {n, 1, 20}]] (* Jean-François Alcover, Nov 27 2017, from Maple *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Kyle Petersen, Aug 09 2016
STATUS
approved