A073165 Triangle T(n,k) read by rows: related to David G. Cantor's sigma function.

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 10, 8, 1, 1, 5, 20, 35, 16, 1, 1, 6, 35, 112, 126, 32, 1, 1, 7, 56, 294, 672, 462, 64, 1, 1, 8, 84, 672, 2772, 4224, 1716, 128, 1, 1, 9, 120, 1386, 9504, 28314, 27456, 6435, 256, 1, 1, 10, 165, 2640, 28314, 151008, 306735, 183040, 24310, 512, 1

Offset: 0

Author

Michael Somos, Jul 24 2002

Keywords

• nonn
• tabl
• easy

Comments

Square array T(n+k,k) read by antidiagonals: number of stars of length k with n branches.

Row n of T(n+k,k) has g.f. (floor(n/2)+1)F(floor(n/2))(1,3/2,5/2,...,(2*floor(n/2)+1)/2;n,n-1,...,n-floor(n/2)+1;2^n*x) (conjecture). [Paul Barry, Jan 23 2009]

Examples

			Triangle rows:
  1;
  1, 1;
  1, 2,  1;
  1, 3,  4,   1;
  1, 4, 10,   8,    1;
  1, 5, 20,  35,   16,    1;
  1, 6, 35, 112,  126,   32,    1;
  1, 7, 56, 294,  672,  462,   64,   1;
  1, 8, 84, 672, 2772, 4224, 1716, 128, 1;
		

Links

• Seiichi Manyama, Rows n = 0..139, 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.
• C. Krattenthaler, A. J. Guttmann and X. G. Viennot, Vicious walkers, friendly walkers and Young tableaux, II: with a wall, arXiv:cond-mat/0006367 [cond-mat.stat-mech], 2000.

Crossrefs

Square array has main diagonal A049505, columns include A001700, A003645, A000356.

Cf. A133112.

Programs

Mathematica

t[n_, k_] := Product[ (n-k+i+j-1) / (i+j-1), {j, 1, k}, {i, 1, j}]; Flatten[ Table[t[n, k], {n, 0, 10}, {k, 0, n}]] (* Jean-François Alcover, May 23 2012, after PARI *)

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)) / prod( i=0, (k-1)\2, binomial(2*k - 2*i - 1, 2*i)))}

PARI

{T(n, k) = if( k<0 || n<0, 0, prod( j=1, k, prod( i=1, j, (n - k + i + j - 1) / (i + j - 1) )))} /* Michael Somos, Oct 16 2006 */

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.

T(n+k, k) = Product_{1<=i<=j<=k} (n+i+j-1)/(i+j-1). - Ralf Stephan, Mar 02 2005

Extensions

Edited by Ralf Stephan, Mar 02 2005