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A107735 Array read by antidiagonals: A(n,k) = Verlinde numbers for quasiparabolic bundles (n >= 3, k >= 0).

%I A107735 #25 Jun 02 2025 00:27:47
%S A107735 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 A107735 43,521,45,61,7,1,1,16,1485,160,1401,76,85,8,1,1,341,171,10569,461,
%U A107735 3101,119,113,9,1,1,32,15565,1088,46649,1112,6021,176,145,10
%N A107735 Array read by antidiagonals: A(n,k) = Verlinde numbers for quasiparabolic bundles (n >= 3, k >= 0).
%D A107735 S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 483.
%F A107735 The reference gives an explicit formula. For odd n this is
%F A107735 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 A107735 For even n use the same formula but replace k by k/2. - _Michel Marcus_, Apr 20 2013
%e A107735 Array begins:
%e A107735 1 1 1 1 1 1 1 1 1 1 ...
%e A107735 1 2 3 4 5 6 7 8 9 10 ...
%e A107735 1 5 13 25 41 61 85 113 ...
%e A107735 1 4 11 24 45 76 119 ...
%e A107735 1 21 141 521 1401 3101 ...
%p A107735 Digits:=100;
%p A107735 A:=proc(n,k) local kp;
%p A107735 if (n mod 2) = 1 then
%p A107735 round( (1/(2*k+1))*add( (-1)^(n*j)*sin( (2*j+1)*Pi/(4*k+2) )^(-n+2), j=0..2*k))
%p A107735 else kp:=k/2;
%p A107735 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 A107735 end;
%t A107735 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 A107735 (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 A107735 Rows: A001844, A006527, A107731; columns: A107732-A107734.
%K A107735 nonn,tabl
%O A107735 3,5
%A A107735 _N. J. A. Sloane_, Jun 10 2005