A055068 Triangular array for David G. Cantor's sigma function.

1, 1, 1, 1, 2, 1, 1, 3, 4, 3, 1, 4, 10, 24, 10, 1, 5, 20, 105, 160, 105, 1, 6, 35, 336, 1260, 3360, 1260, 1, 7, 56, 882, 6720, 48510, 80640, 48510, 1, 8, 84, 2016, 27720, 443520, 2162160, 6209280, 2162160, 1, 9, 120, 4158, 95040, 2972970, 34594560, 312161850

Offset

0,5

Links

Paolo Xausa, Table of n, a(n) for n = 0..3320 (rows 0..80 of triangle, flattened).

D. G. Cantor, On the analogue of the division polynomials for hyperelliptic curves, J. Reine Angew. Math. (Crelle's J.) 447 (1994), pp. 91-145.

Formula

T(n, k)*T(n-2, k-1)-2*T(n-1, k-1)*T(n-1, k)+T(n, k-1)*T(n-2, k)=0.

Example

Triangle rows:
  1;
  1,1;
  1,2,1;
  1,3,4,3;
  1,4,10,24,10;
  1,5,20,105,160,105;
  ...

Mathematica

A055068[n_, k_] := Product[Binomial[n - # + i, 2*(i - #) + 1], {i, 1, k, 2}] & [Mod[k, 2]];
Table[A055068[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Aug 13 2025, after PARI *)

Prog

(PARI) {T(n, k) = if( k<0 || k>n, 0, prod( i=1, (k+1)\2, binomial(n + 2*i - 1 - k%2, 4*i - 1 - k%2*2)))}

Crossrefs

Sequence in context: A025564 A052265 A306565 * A237498 A319516 A015138

Adjacent sequences: A055065 A055066 A055067 * A055069 A055070 A055071

Keyword

nonn,tabl,easy

Author

Michael Somos, Jun 12 2000

Status

approved