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 A358612 #48 Jun 21 2024 14:17:52
%S A358612 1,1,1,3,1,1,5,2,1,7,6,1,1,9,4,1,11,11,2,1,13,15,3,1,15,25,10,1,1,17,
%T A358612 8,1,19,21,4,1,21,28,6,1,23,44,19,2,1,25,39,9,1,27,58,27,3,1,29,68,34,
%U A358612 4,1,31,90,65,15,1,1,33,16,1,35,41,8,1,37,54,12,1
%N A358612 Irregular table T(n, k), n >= 0, k > 0, read by rows of extended (due to binary expansion of n) Stirling numbers of the second kind.
%C A358612 Row n length is A000120(n) + 2.
%F A358612 T(n, 1) = 1 for n > 0 with T(0, 1) = T(0, 2) = 1.
%F A358612 T(2n+1, k) = k*T(n, k) + T(n, k-1) for n >= 0, k > 1.
%F A358612 T(2n, k) = k*T(n, k) + T(n, k-1) - (T(2n, k-1) + T(n, k-1))/(k-1) for n > 0, k > 1.
%F A358612 T(2^n - 1, k) = Stirling2(n+2, k) for n >= 0, k > 0.
%F A358612 T(n, 2) = 2n+1 for n >= 0.
%F A358612 Conjectured formulas: (Start)
%F A358612 T(n, A000120(n) + 2) = A341392(n) for n >= 0.
%F A358612 Sum_{i=1..wt(k) + 2} i!*i^m*T(k, i)*(-1)^(wt(k) - i + 2) = A329369(2^m*(2k+1)) for m >= 0, k >= 0 where wt(n) = A000120(n). (End)
%F A358612 Conjecture: T(n, k) = (k-1)^g(n)*T(h(n), k-1) + k^(g(n)+1)*T(h(n), k) for n > 0, k > 1 with T(n, 1) = T(0, 2) = 1 where g(n) = A007814(n) and where h(n) = A025480(n-1). - _Mikhail Kurkov_, Jun 21 2024
%e A358612 Irregular table begins:
%e A358612 1, 1;
%e A358612 1, 3, 1;
%e A358612 1, 5, 2;
%e A358612 1, 7, 6, 1;
%e A358612 1, 9, 4;
%e A358612 1, 11, 11, 2;
%e A358612 1, 13, 15, 3;
%e A358612 1, 15, 25, 10, 1;
%e A358612 1, 17, 8;
%e A358612 1, 19, 21, 4;
%e A358612 1, 21, 28, 6;
%e A358612 1, 23, 44, 19, 2;
%e A358612 1, 25, 39, 9;
%e A358612 1, 27, 58, 27, 3;
%e A358612 1, 29, 68, 34, 4;
%e A358612 1, 31, 90, 65, 15, 1;
%o A358612 (PARI) T(n, k)=if(n==0 || k==1, (n==0 && k<3) + (k==1 && n>0), k*T(n\2, k) + T(n\2, k-1) - if(n%2==0, (T(n, k-1) + T(n\2,k-1))/(k-1)))
%o A358612 (PARI) row(n) = my(A, v1, v2); v1 = [1, 1]; if(n == 0, v1, forstep(i=logint(n, 2), 0, -1, A = bittest(n, i); v2 = vector(#v1+A, i, 0); v2[1] = 1; for(j=2, #v2, v2[j] = j*if(j==#v1+1, 0, v1[j]) + v1[j-1] - if(A, 0, (v2[j-1] + v1[j-1])/(j-1))); v1 = v2); v1) \\ _Mikhail Kurkov_, Apr 30 2024
%Y A358612 Cf. A000120, A007814, A008277, A025480, A329369, A341392, A357990, A373183.
%Y A358612 Similar tables: A358631.
%K A358612 nonn,base,tabf
%O A358612 0,4
%A A358612 _Mikhail Kurkov_, Nov 23 2022