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A073165
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Triangle T(n,k) read by rows: related to David G. Cantor's sigma function.
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6
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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
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OFFSET
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0,5
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COMMENTS
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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). [From Paul Barry, Jan 23 2009]
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REFERENCES
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David G. Cantor, On the analogue of the division polynomials for hyperelliptic curves, J. Reine Angew. Math. 447 (1994), 91-145.
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LINKS
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Table of n, a(n) for n=0..62.
C. Krattenthaler, A. J. Guttmann and X. G. Viennot, Vicious walkers, friendly walkers and Young tableaux, II: with a wall
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FORMULA
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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) = Prod[1<=i<=j<=k, (n+i+j-1)/(i+j-1) ]. - Ralf Stephan
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EXAMPLE
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Triangle rows: 1; 1,1; 1,2,1; 1,3,4,1; 1,4,10,8,1; 1,5,20,35,16,1; ...
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MATHEMATICA
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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 *)
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PROG
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(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 */
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CROSSREFS
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Square array has main diagonal A049505, columns include A001700, A003645, A000356.
Sequence in context: A175105 A162717 A122175 * A137153 A063841 A137596
Adjacent sequences: A073162 A073163 A073164 * A073166 A073167 A073168
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KEYWORD
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nonn,tabl,easy
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AUTHOR
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Michael Somos, Jul 24 2002
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EXTENSIONS
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Edited by Ralf Stephan, Mar 02 2005
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STATUS
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approved
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