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 A033878 #25 Jul 08 2025 19:58:12 %S A033878 1,1,1,1,3,2,1,5,10,6,1,7,22,38,22,1,9,38,98,158,90,1,11,58,194,450, %T A033878 698,394,1,13,82,334,978,2126,3218,1806,1,15,110,526,1838,4942,10286, %U A033878 15310,8558,1,17,142,778,3142,9922,25150,50746,74614,41586 %N A033878 Triangular array associated with Schroeder numbers. %C A033878 Transpose of triangular array A132372. - _Michel Marcus_, May 02 2015 %H A033878 E. Pergola and R. A. Sulanke, <a href="https://cs.uwaterloo.ca/journals/JIS/PergolaSulanke/">Schroeder Triangles, Paths and Parallelogram Polyominoes</a>, J. Integer Sequences, 1 (1998), #98.1.7. %e A033878 This triangle reads: %e A033878 1 %e A033878 1 1 %e A033878 1 3 2 %e A033878 1 5 10 6 %e A033878 1 7 22 38 22 %e A033878 1 9 38 98 158 90 %e A033878 1 11 58 194 450 698 394 %e A033878 1 13 82 334 978 2126 3218 1806 %e A033878 1 15 110 526 1838 4942 10286 15310 558 %e A033878 1 17 142 778 3142 9922 25150 50746 74614 41586 %o A033878 (PARI) lgs(n) = if( n<1, 1, sum( k=0, n, 2^k * binomial( n, k) * binomial( n, k-1)) / n) /* A006318 */ %o A033878 T(n, k) = if (k>n, 0, if (k==0, 1, if (n==0, 1, if ((n==k), lgs(n-1), T(n,k-1) + T(n-1,k-1) + T(n-1,k))))); %o A033878 tabl(nn) = {for (n=0, nn, for (m=0, n, print1(T(n, m), ", ");); print(););} \\ _Michel Marcus_, May 02 2015 %o A033878 (Python) %o A033878 from functools import cache %o A033878 @cache %o A033878 def T(n: int, k: int) -> int: %o A033878 if n < 0: return 0 %o A033878 if k == 0: return 1 %o A033878 if n == k: return sum(T(n-1, k) for k in range(n)) %o A033878 return T(n, k-1) + T(n-1, k-1) + T(n-1, k) %o A033878 for n in range(10): %o A033878 print([T(n, k) for k in range(n+1)]) # _Peter Luschny_, Dec 26 2024 %Y A033878 Cf. A132372, A006318. %K A033878 nonn,tabl,easy %O A033878 0,5 %A A033878 _N. J. A. Sloane_ %E A033878 Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 23 2003