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%I #23 Jan 14 2014 10:04:47
%S 1,1,1,1,2,1,1,5,3,1,1,4,13,4,1,1,21,11,25,5,1,1,8,141,24,41,6,1,1,85,
%T 43,521,45,61,7,1,1,16,1485,160,1401,76,85,8,1,1,341,171,10569,461,
%U 3101,119,113,9,1,1,32,15565,1088,46649,1112,6021,176,145,10
%N Array read by antidiagonals: A(n,k) = Verlinde numbers for quasiparabolic bundles (n >= 3, k >= 0)
%D S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 483.
%F The reference gives an explicit formula. For odd n this is
%F A(n,k) = (1/(2*k+1))*sum( (-1)^(n*j)*sin( (2*j+1)*Pi/(4*k+2) )^(-n+2), j=0..2*k). - _N. J. A. Sloane_, Apr 20 2013.
%F For even n use the same formula but replace k by k/2. - _Michel Marcus_, Apr 20 2013
%e Array begins:
%e 1 1 1 1 1 1 1 1 1 1 ...
%e 1 2 3 4 5 6 7 8 9 10 ...
%e 1 5 13 25 41 61 85 113 ...
%e 1 4 11 24 45 76 119 ...
%e 1 21 141 521 1401 3101 ...
%p Digits:=100;
%p A:=proc(n,k) local kp;
%p if (n mod 2) = 1 then
%p round( (1/(2*k+1))*add( (-1)^(n*j)*sin( (2*j+1)*Pi/(4*k+2) )^(-n+2), j=0..2*k))
%p else kp:=k/2;
%p round( (1/(2*kp+1))*add( (-1)^(n*j)*sin( (2*j+1)*Pi/(4*kp+2) )^(-n+2), j=0..2*kp)); fi;
%p end;
%t t[n_, k_] := With[{kp = If[!Divisible[n, 2], k, k/2]}, Round[1/(2*kp+1)*Sum[(-1)^(n*j)*Sin[(2*j+1)*Pi/(4*kp+2)]^(-n+2), {j, 0, 2*kp}]]]; Table[t[n-k, k], {n, 3, 13}, {k, 0, n-3}] // Flatten (* _Jean-François Alcover_, Jan 14 2014, after _Michel Marcus_ *)
%o (PARI) t(n, k) = {if (! (n % 2), k = k/2); return (round((1/(2*k+1))*sum(j=0, 2*k,(-1)^(n*j)*sin((2*j+1)*Pi/(4*k+2))^(-n+2))));} \\ _Michel Marcus_, Apr 20 2013
%Y Rows: A001844, A006527, A107731; columns: A107732-A107734.
%K nonn,tabl
%O 3,5
%A _N. J. A. Sloane_, Jun 10 2005
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