A107735 Array read by antidiagonals: A(n,k) = Verlinde numbers for quasiparabolic bundles (n >= 3, k >= 0).
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, 43, 521, 45, 61, 7, 1, 1, 16, 1485, 160, 1401, 76, 85, 8, 1, 1, 341, 171, 10569, 461, 3101, 119, 113, 9, 1, 1, 32, 15565, 1088, 46649, 1112, 6021, 176, 145, 10
Offset: 3
Examples
Array begins: 1 1 1 1 1 1 1 1 1 1 ... 1 2 3 4 5 6 7 8 9 10 ... 1 5 13 25 41 61 85 113 ... 1 4 11 24 45 76 119 ... 1 21 141 521 1401 3101 ...
References
- S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 483.
Programs
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Maple
Digits:=100; A:=proc(n,k) local kp; if (n mod 2) = 1 then round( (1/(2*k+1))*add( (-1)^(n*j)*sin( (2*j+1)*Pi/(4*k+2) )^(-n+2), j=0..2*k)) else kp:=k/2; round( (1/(2*kp+1))*add( (-1)^(n*j)*sin( (2*j+1)*Pi/(4*kp+2) )^(-n+2), j=0..2*kp)); fi; end;
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Mathematica
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 *) -
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
Formula
The reference gives an explicit formula. For odd n this is
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.
For even n use the same formula but replace k by k/2. - Michel Marcus, Apr 20 2013