A373183 - OEIS

%I #17 Jun 28 2024 23:14:45

%S 1,0,1,1,2,0,0,1,3,4,0,1,2,1,3,3,0,0,0,1,7,8,0,3,4,3,8,6,0,0,1,2,7,15,

%T 9,0,1,3,3,1,4,6,4,0,0,0,0,1,15,16,0,7,8,7,18,12,0,0,3,4,17,34,18,0,3,

%U 8,6,3,11,15,8,0,0,0,1,2,31,57,27,0,7,15

%N Irregular table 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) = x*(R(n, x+1) - R(n, x)) for n > 0 with R(0, x) = x.

%C Row n length is A000120(n) + 1.

%F Conjectured formulas: (Start)

%F R(2n, x) = R(n, x) + R(n - 2^f(n), x) + R(2n - 2^f(n), x) where f(n) = A007814(n) (see A329369).

%F b(2^m*n + q) = Sum_{i=A001511(n+1)..A000120(n)+1} T(n, i)*b(2^m*(2^(i-1)-1) + q) for n >= 0, m >= 0, q >= 0 where b(n) = A329369(n). Note that this formula is recursive for n != 2^k - 1.

%F R(n, x) = c(n, x)

%F where c(2^k - 1, x) = x^(k+1) for k >= 0,

%F c(n, x) = Sum_{i=0..s(n)} p(n, s(n)-i)/i!*Sum_{j=0..i} (s(n)-j+1)^A279209(n)*binomial(i, j)*(-1)^j,

%F p(n, k) = Sum_{i=0..k} c(t(n) + (2^i - 1)*A062383(t(n)), x)*L(s(n), k, i) for 0 <= k < s(n) with p(n, s(n)) = c(t(n) + (2^s(n) - 1)*A062383(t(n)), x),

%F s(n) = A090996(n), t(n) = A087734(n),

%F L(n, k, m) are some integer coefficients defined for n > 0, 0 <= k < n, 0 <= m <= k that can be represented as (n - k)!*W(n-m, k-m, m+1)

%F and where W(n, k, m) = (k+m)*W(n-1, k, m) + (n-k)*W(n-1, k-1, m) + [m > 1]*W(n, k, m-1) for 0 <= k < n, m > 0 with W(0, 0, m) = 1, W(n, k, m) = 0 for n < 0 or k < 0.

%F In particular, W(n, k, 1) = A173018(n, k), W(n, k, 2) = A062253(n, k), W(n, k, 3) = A062254(n, k) and W(n, k, 4) = A062255(n, k).

%F Here s(n), t(n) and A279209(n) are unique integer sequences such that n can be represented as t(n) + (2^s(n) - 1)*A062383(t(n))*2^A279209(n) where t(n) is minimal. (End)

%F Conjectures from _Mikhail Kurkov_, Jun 19 2024: (Start)

%F T(n, k) = d(n, 1, A000120(n) - k + 2) where d(n, m, k) = (m+1)^g(n)*d(h(n), m+1, k) - m^(g(n)+1)*d(h(n), m, k-1) for n > 0, m > 0, k > 0 with d(n, m, 0) = 0 for n >= 0, m > 0, d(0, m, k) = [k <= m]*abs(Stirling1(m, m-k+1)) for m > 0, k > 0, g(n) = A290255(n) and where h(n) = A053645(n). In particular, d(n, 1, 1) = A341392(n).

%F Sum_{i=A001511(n+1)..wt(n)+k} d(n, k, wt(n)-i+k+1)*A329369(2^m*(2^(i-1)-1) + q) = k!*A357990(2^m*n + q, k) for n >= 0, k > 0, m >= 0, q >= 0 where wt(n) = A000120(n).

%F If we change R(0, x) to Sum_{i=1..m} abs(Stirling1(m, i))*x^i, then for resulting irregular table U(n, k, m) we have U(n, k, m) = d(n, m, A000120(n) - k + m + 1).

%F T(n, k) = (-1)^(wt(n)-k+1)*Sum_{i=1..wt(n)-k+3} Stirling1(wt(n)-i+3, k+1)*A358612(n, wt(n)-i+3) for n >= 0, k > 0 where wt(n) = A000120(n). (End)

%e Irregular table begins:

%e   1;

%e   0,  1;

%e   1,  2;

%e   0,  0, 1;

%e   3,  4;

%e   0,  1, 2;

%e   1,  3, 3;

%e   0,  0, 0, 1;

%e   7,  8;

%e   0,  3, 4;

%e   3,  8, 6

%e   0,  0, 1, 2

%e   7, 15, 9;

%e   0,  1, 3, 3;

%e   1,  4, 6, 4;

%e   0,  0, 0, 0, 1;

%o (PARI) row(n) = my(x = 'x, A = x); forstep(i=if(n == 0, -1, logint(n, 2)), 0, -1, A = if(bittest(n, i), x*A, x*(subst(A, x, x+1) - A))); Vecrev(A/x)

%Y Cf. A000120, A001511, A007814, A053645, A062253, A062254, A062255, A062383, A087734, A090996, A173018, A279209, A290255, A329369, A341392, A357990, A358612.

%K nonn,base,tabf

%O 0,5

%A _Mikhail Kurkov_, May 27 2024