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A275787
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Number of cells in the two-sided Coxeter complex of type B_n.
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2
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1, 5, 41, 509, 8469, 176217, 4400325, 128203049, 4268957449, 159922273421, 6656731517249, 304797275277365, 15224868078068845, 823874409422614577, 48012621942105876301, 2997884066292303095889, 199666128081901473290833, 14129411123649333432720277, 1058688691179737704258634521, 83732563305101190468369022317, 6971039973751002759723517967941
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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EXAMPLE
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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]
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MAPLE
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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);
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MATHEMATICA
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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 *)
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CROSSREFS
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Cf. A120733 gives the number of cells for type A_n.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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