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

0,5

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]

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.

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

Example

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;

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 *)

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)) / 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 */

Crossrefs

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

Cf. A133112.

Sequence in context: A175105 A162717 A122175 * A137153 A340814 A063841

Adjacent sequences: A073162 A073163 A073164 * A073166 A073167 A073168

Keyword

nonn,tabl,easy

Author

Michael Somos, Jul 24 2002

Extensions

Edited by Ralf Stephan, Mar 02 2005

Status

approved