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A379817 Irregular table T(n, k), n >= 0, k >= 0, read by rows such that T(n,k) = f(n,k)/f(2^k-1,k) where f(n,k) is defined in Comments.

%I A379817 #6 Jan 14 2025 16:11:44
%S A379817 1,1,1,3,2,1,3,1,7,4,3,7,2,7,12,3,1,7,6,1,15,8,7,15,4,17,26,6,3,17,13,
%T A379817 2,31,42,9,7,31,21,3,15,50,30,4,1,15,25,10,1,31,16,15,31,8,37,54,12,7,
%U A379817 37,27,4,69,88,18,17,69,44,6,37,112,63,8,3,37,56
%N A379817 Irregular table T(n, k), n >= 0, k >= 0, read by rows such that T(n,k) = f(n,k)/f(2^k-1,k) where f(n,k) is defined in Comments.
%C A379817 Here f(n,k) = b(2^k*(2n+1)) - Sum_{j=1..k} b(2^(j-1)*(2n+1))*R(k,j) for n >= 0, k >= 0 where b(n) = A329369(n) and where R(k,j) is the unique solution to b(2^k*(2^i-1)) = Sum_{j=1..k} b(2^(j-1)*(2^i-1))*R(k,j) for k > 0, 1 <= i <= k.
%C A379817 Row n length is A000120(n) + 1.
%F A379817 Conjectures: (Start)
%F A379817 f(2^k-1,k) = ((k+1)!)^2 for k >= 0.
%F A379817 R(k,j) = -Stirling1(k+2, j+1) for k > 0, 1 <= j <= k.
%F A379817 T(2^n-1, k) = Stirling2(n+1, k+1) for n >= 0, 0 <= k <= n.
%F A379817 T(n,k) = c(n,wt(n)-k) for n >= 0, 0 <= k <= wt(n) where c(2n+1,k) = c(n,k) + (wt(n)-k+2)*c(n,k-1), c(2n,k) = (wt(n)-k+1)*c(2n+1,k) for n > 0, k > 0 with c(n,0) = A341392(n) for n >= 0, c(0,k) = 0 for k > 0 and where wt(n) = A000120(n). (End)
%e A379817 Irregular table begins:
%e A379817    1;
%e A379817    1,  1;
%e A379817    3,  2;
%e A379817    1,  3,  1;
%e A379817    7,  4;
%e A379817    3,  7,  2;
%e A379817    7, 12,  3;
%e A379817    1,  7,  6,  1;
%e A379817   15,  8;
%e A379817    7, 15,  4;
%e A379817   17, 26,  6;
%e A379817    3, 17, 13,  2;
%e A379817   31, 42,  9;
%e A379817    7, 31, 21,  3;
%e A379817   15, 50, 30,  4;
%e A379817    1, 15, 25, 10, 1;
%o A379817 (PARI) upto(n) = my(A, v1); v1 = vector(n+1, i, 0); v1[1] = 1; for(i=1, n, v1[i+1] = v1[i\2+1] + if(i%2, 0, A = 1 << valuation(i/2, 2); v1[i/2-A+1] + v1[i-A+1])); v1 \\ from A329369
%o A379817 R(k) = my(v1, M1, M2); v1 = upto(2^k*(2^k-1)); M1 = matrix(k, k, i, j, v1[2^(j-1)*(2^i-1)+1]); M2 = matrix(k, 1, i, j, v1[2^k*(2^i-1)+1]); M1 = matsolve(M1, M2)
%o A379817 row(n) = my(A = hammingweight(n), v1, v2, v3); v1 = upto(2^A*(2*n+1)); v2 = vector(A, i, R(i)); v3 = vector(A, i, (v1[2^i*(2*n+1)+1] - sum(j=1, i, v1[2^(j-1)*(2*n+1)+1]*v2[i][j,1]))/(v1[2^i*(2*(2^i-1)+1)+1] - sum(j=1, i, v1[2^(j-1)*(2*(2^i-1)+1)+1]*v2[i][j,1]))); concat(v1[n+1], v3)
%Y A379817 Cf. A000120, A341392, A329369.
%K A379817 nonn,base,tabf
%O A379817 0,4
%A A379817 _Mikhail Kurkov_, Jan 03 2025