This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A073165 #34 Aug 11 2025 06:30:11
%S A073165 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,
%T A073165 7,56,294,672,462,64,1,1,8,84,672,2772,4224,1716,128,1,1,9,120,1386,
%U A073165 9504,28314,27456,6435,256,1,1,10,165,2640,28314,151008,306735,183040,24310,512,1
%N A073165 Triangle T(n,k) read by rows: related to David G. Cantor's sigma function.
%C A073165 Square array T(n+k,k) read by antidiagonals: number of stars of length k with n branches.
%C A073165 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]
%H A073165 Seiichi Manyama, <a href="/A073165/b073165.txt">Rows n = 0..139, flattened</a>
%H A073165 D. G. Cantor, <a href="https://gdz.sub.uni-goettingen.de/id/PPN243919689_0447">On the analogue of the division polynomials for hyperelliptic curves</a>, J. Reine Angew. Math. (Crelle's J.) 447 (1994), pp. 91-145.
%H A073165 C. Krattenthaler, A. J. Guttmann and X. G. Viennot, <a href="http://arXiv.org/abs/cond-mat/0006367">Vicious walkers, friendly walkers and Young tableaux, II: with a wall</a>, arXiv:cond-mat/0006367 [cond-mat.stat-mech], 2000.
%F A073165 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.
%F A073165 T(n+k, k) = Product_{1<=i<=j<=k} (n+i+j-1)/(i+j-1). - _Ralf Stephan_, Mar 02 2005
%e A073165 Triangle rows:
%e A073165 1;
%e A073165 1, 1;
%e A073165 1, 2, 1;
%e A073165 1, 3, 4, 1;
%e A073165 1, 4, 10, 8, 1;
%e A073165 1, 5, 20, 35, 16, 1;
%e A073165 1, 6, 35, 112, 126, 32, 1;
%e A073165 1, 7, 56, 294, 672, 462, 64, 1;
%e A073165 1, 8, 84, 672, 2772, 4224, 1716, 128, 1;
%t A073165 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 *)
%o A073165 (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)))}
%o A073165 (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 */
%Y A073165 Square array has main diagonal A049505, columns include A001700, A003645, A000356.
%Y A073165 Cf. A133112.
%K A073165 nonn,tabl,easy
%O A073165 0,5
%A A073165 _Michael Somos_, Jul 24 2002
%E A073165 Edited by _Ralf Stephan_, Mar 02 2005