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 A116445 #14 Aug 17 2022 16:21:07 %S A116445 1,1,1,1,3,1,1,3,5,1,1,3,8,7,1,1,3,8,16,9,1,1,3,8,20,27,11,1,1,3,8,20, %T A116445 43,41,13,1,1,3,8,20,48,81,58,15,1,1,3,8,20,48,106,138,78,17,1,1,3,8, %U A116445 20,48,112,213,218,101,19,1 %N A116445 Array read by antidiagonals: the binomial transform of the sequence (1,2,..n,0,0,0..) in row n. %C A116445 Create an array by rows: (binomial transforms of 1,0,0,0,...; 1,2,0,0,0,...; 1,2,3,0,0,0,...; etc.). Antidiagonals of the array become rows of the triangle. %e A116445 First few rows of the array: %e A116445 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... %e A116445 1, 3, 5, 7, 9, 11, 13, 15, 17, ... %e A116445 1, 3, 8, 16, 27, 41, 58, 78, 101, ... A104249 %e A116445 1, 3, 8, 20, 43, 81, 138, 218, ... A139488 %e A116445 1, 3, 8, 20, 48, 106, 213, ... %e A116445 1, 3, 8, 20, 48, 112, 249, ... %e A116445 ... %e A116445 Diagonals converge to A001792, binomial transform of (1,2,3,...); and the first few rows of the triangle created by reading upwards antidiagonals are: %e A116445 1 %e A116445 1, 1; %e A116445 1, 3, 1; %e A116445 1, 3, 5, 1; %e A116445 1, 3, 8, 7, 1; %e A116445 1, 3, 8, 16, 9, 1; %e A116445 1, 3, 8, 20, 27, 22, 1; %e A116445 ... %e A116445 a(4), a(5), a(6) = 1, 3, 1 = antidiagonals of the array becoming row three of the triangle. %p A116445 A116445 := proc(n,k) %p A116445 local a,i ; %p A116445 a := 0 ; %p A116445 for i from 0 to n do %p A116445 a := a+binomial(k,i)*(i+1) ; %p A116445 end do: %p A116445 a ; %p A116445 end proc: %p A116445 seq(seq(A116445(d-k,k),k=0..d),d=0..12) ; # _R. J. Mathar_, Aug 17 2022 %Y A116445 Cf. A001629 (antidiagonal sums), A104249. %K A116445 nonn,tabl,easy %O A116445 1,5 %A A116445 _Gary W. Adamson_, Feb 15 2006 %E A116445 Detailed NAME by _R. J. Mathar_, Aug 17 2022