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 A358631 #37 Nov 07 2024 11:13:28
%S A358631 1,1,2,3,1,4,5,1,6,11,6,1,6,7,1,12,20,9,1,18,26,9,1,24,50,35,10,1,8,9,
%T A358631 1,18,29,12,1,30,41,12,1,48,94,59,14,1,36,47,12,1,72,130,71,14,1,96,
%U A358631 154,71,14,1,120,274,225,85,15,1,10,11,1,24,38,15,1,42
%N A358631 Irregular table T(n, k), n >= 0, k > 0, read by rows of extended (due to binary expansion of n) Stirling numbers of the first kind.
%C A358631 Row n length is A000120(n) + 2.
%F A358631 T(n, k) = P(n, wt(n) - k + 3) for n >= 0, 0 < k <= wt(n) + 2 where wt(n) = A000120(n).
%F A358631 P(n, 1) = 1 for n > 0 with P(0, 1) = P(0, 2) = 1.
%F A358631 P(n, k) = (A000120(q(n)) + 2)*P(q(n), k-1)*(A290255(n) + 1) + P(s(q(n)), k) for n > 0, k > 1 where q(n) = A053645(n) and where s(n) = n + [A063250(n) > 0]*2^(A063250(n) - 1).
%F A358631 T(2^n - 1, k) = abs(Stirling1(n+2, k)) for n >= 0, k > 0.
%F A358631 Conjectures: (Start)
%F A358631 T(n, 1) = (A000120(n) + 1)!*A347205(n) for n >= 0.
%F A358631 Sum_{k=1..A000120(n) + 2} T(n, k)*(-1)^k = 0 for n >= 0.
%F A358631 Sum_{k=0..2^n - 1} Sum_{j=1..A000120(k) + 2} T(k, j) = 2*A052852(n+1) for n >= 0.
%F A358631 Sum_{i=1..wt(k) + 2} m^(i-1)*T(k, i) = (wt(k) + 1)!*A347205(2^m*(2k+1)) for m >= 0, k >= 0 where wt(n) = A000120(n). (End)
%e A358631 Irregular table begins:
%e A358631 1, 1;
%e A358631 2, 3, 1;
%e A358631 4, 5, 1;
%e A358631 6, 11, 6, 1;
%e A358631 6, 7, 1;
%e A358631 12, 20, 9, 1;
%e A358631 18, 26, 9, 1;
%e A358631 24, 50, 35, 10, 1;
%e A358631 8, 9, 1;
%e A358631 18, 29, 12, 1;
%e A358631 30, 41, 12, 1;
%e A358631 48, 94, 59, 14, 1;
%e A358631 36, 47, 12, 1;
%e A358631 72, 130, 71, 14, 1;
%e A358631 96, 154, 71, 14, 1;
%e A358631 120, 274, 225, 85, 15, 1;
%o A358631 (PARI) b1(n)=if(n>0, my(A=n - 2^logint(n, 2)); if(A>0, logint(A, 2) + 1))
%o A358631 b2(n)=if(n>0, my(A=b1(3*2^logint(n, 2) - n - 1)); n + if(A>0, 2^(A-1)))
%o A358631 P(n,k)=if(n==0 || k==1, (n==0 && k<3) + (k==1 && n>0), my(L=logint(n, 2), A=n - 2^L); (hammingweight(A) + 2)*P(A, k-1)*(L - b1(n) + 1) + P(b2(A), k))
%o A358631 T(n, k)=my(A=hammingweight(n)); if(k<=(A + 2), P(n, A - k + 3))
%Y A358631 Cf. A000120, A053645, A052852, A063250, A132393, A290255, A347205.
%Y A358631 Similar tables: A358612.
%K A358631 nonn,base,tabf
%O A358631 0,3
%A A358631 _Mikhail Kurkov_, Nov 24 2022
%E A358631 Offset corrected by _Mikhail Kurkov_, Nov 07 2024