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A073165 Triangle T(n,k) read by rows: related to David G. Cantor's sigma function. 8
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified November 29 11:46 EST 2021. Contains 349416 sequences. (Running on oeis4.)