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A380971 Irregular triangle T(n, k), n >= 0, k > 0, read by rows with row polynomials R(n, x) such that R(2n+1, x) = x*R(n, x) for n >= 0, R(2n,x) = wt(n)*x*((x+1)^wt(n) - x^wt(n)) + Sum_{k=1..wt(n)} k*x^k*T(n,k) for n > 0 with R(0,x) = 0 where wt(n) = A000120(n).

%I A380971 #5 Feb 12 2025 12:59:14
%S A380971 1,2,0,1,2,4,3,0,2,2,6,0,0,1,4,12,0,2,4,3,9,9,4,0,3,2,8,0,0,2,4,16,0,
%T A380971 2,6,3,9,12,0,0,0,1,6,28,0,4,12,3,13,21,0,0,2,4,6,27,36,0,3,9,9,4,16,
%U A380971 24,16,5,0,4,2,10,0,0,3,4,20,0,2,8,3,9,15,0
%N A380971 Irregular triangle T(n, k), n >= 0, k > 0, read by rows with row polynomials R(n, x) such that R(2n+1, x) = x*R(n, x) for n >= 0, R(2n,x) = wt(n)*x*((x+1)^wt(n) - x^wt(n)) + Sum_{k=1..wt(n)} k*x^k*T(n,k) for n > 0 with R(0,x) = 0 where wt(n) = A000120(n).
%C A380971 Row n length is A000120(n) except for n = 2^k - 1 which are empty rows.
%F A380971 Conjectures: (Start)
%F A380971 b(2^m*(2k+1)) = b(2^m*(2^wt(k)-1)) + Sum_{i=1..wt(k)} (i+1)^m*T(k,i)*(-1)^(wt(k)-i) for m >= 0, k >= 0 where b(n) = A380944(n) and where wt(n) = A000120(n). Note that this formula is recursive for k != 2^q - 1. We can also use b(2^m*(2^n-1)) = (n+1)^m - n*n!*c(m,n+1) for n >= 0, m >= 0 where c(n,k) = Sum_{i=0..n-k} Stirling2(k+i,k) for n >= 0, k >= 0.
%F A380971 A380179(n,k) = Sum_{i=0..2^(n+1)-1, [wt(i)<=(k+1)]*T(i,k+1)*(-1)^(wt(i)-k+1) for 0 <= k < n. (End)
%e A380971 Irregular triangle begins:
%e A380971   -
%e A380971   -
%e A380971   1;
%e A380971   -
%e A380971   2;
%e A380971   0,  1;
%e A380971   2,  4;
%e A380971   -
%e A380971   3;
%e A380971   0,  2;
%e A380971   2,  6;
%e A380971   0,  0, 1;
%e A380971   4, 12;
%e A380971   0,  2, 4;
%e A380971   3,  9, 9;
%e A380971   -
%o A380971 (PARI) row(n) = if(n==0, [], my(x = 'x, A = 0, B = 0); forstep(i=logint(n, 2), 0, -1, A = if(bittest(n, i), B++; x*A, B*x*((x+1)^B - x^B) + sum(k=1, B, k*x^k*polcoeff(A, k, x)))); Vecrev(A/x))
%Y A380971 Cf. A000120, A373183, A380179, A380944.
%K A380971 nonn,base,tabf
%O A380971 0,2
%A A380971 _Mikhail Kurkov_, Feb 10 2025