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A346083 Triangle, read by rows, defined by recurrence: T(n,k) = T(n-1,k-1) + (-1)^k * (2 * k + 1) * T(n-1,k) for 0 < k < n with initial values T(n,0) = T(n,n) = 1 for n >= 0 and T(i,j) = 0 if j < 0 or j > i.

%I A346083 #16 Aug 20 2021 17:46:40
%S A346083 1,1,1,1,-2,1,1,7,3,1,1,-20,22,-4,1,1,61,90,50,5,1,1,-182,511,-260,95,
%T A346083 -6,1,1,547,2373,2331,595,161,7,1,1,-1640,12412,-13944,7686,-1176,252,
%U A346083 -8,1,1,4921,60420,110020,55230,20622,2100,372,9,1,1,-14762,307021,-709720,607090,-171612,47922,-3480,525,-10,1
%N A346083 Triangle, read by rows, defined by recurrence: T(n,k) = T(n-1,k-1) + (-1)^k * (2 * k + 1) * T(n-1,k) for 0 < k < n with initial values T(n,0) = T(n,n) = 1 for n >= 0 and T(i,j) = 0 if j < 0 or j > i.
%F A346083 G.f. of column k >= 0: col(t,k) = Sum_{n >= k} T(n,k) * t^n = t^k / (Product_{i=0..k} (1 - (-1)^i * (2 * i + 1) * t)), i.e., col(t,k) = col(t,k-1) * t / (1 - (-1)^k * (2 * k + 1) * t) for k > 0.
%F A346083 Matrix inverse M = T^(-1) has row polynomials p(n,x) = Sum_{k=0..n} M(n,k) * x^k = Product_{i=1..n} (x + (-1)^i * (2 * i - 1)) for n >= 0 and empty product 1, i.e., p(n,x) = p(n-1,x) * (x + (-1)^n * (2 * n - 1)) for n > 0 with initial value p(0,x) = 1.
%F A346083 Conjecture: E.g.f. of column k >= 0: Sum_{n >= k} T(n,k) * t^n / (n!) = (Sum_{i=0..k} (-1)^(i * (i + 1 ) / 2) * binomial(k,floor((k - i) / 2)) * exp((-1)^i * (2 * i + 1) * t)) * (-1)^(k * (k - 1) / 2) / (4^k * (k!)), i.e., T(n,k) = (Sum_{i=0..k} (-1)^(i * (i + 1) / 2) * binomial(k,floor((k - i) / 2)) * ((-1)^i * (2 * i + 1))^n) * (-1)^(k * (k - 1) / 2) / (4^k * (k!)) for 0 <= k <= n.
%F A346083 Conjecture: E.g.f. of column k >= 0: Sum_{n >= k} T(n,k) * t^n / (n!) = exp(t) * (exp(4*t) - 1)^k / (4^k * (k!) * exp(4*t*floor((k+1)/2))), i.e., T(n,k) = (Sum_{i=0..k} (-1)^i * binomial(k,i) *(1 + 4*i - 4*floor((k+1)/2))^n) * (-1)^k / (4^k * (k!)) for 0 <= k <= n. Proved by Burkhard Hackmann and Werner Schulte (distinction of two cases: odd k, even k). - _Werner Schulte_, Aug 03 2021
%e A346083 The triangle T(n,k) for 0 <= k <= n starts:
%e A346083 n\k :  0      1      2       3      4      5     6    7  8  9
%e A346083 =============================================================
%e A346083   0 :  1
%e A346083   1 :  1      1
%e A346083   2 :  1     -2      1
%e A346083   3 :  1      7      3       1
%e A346083   4 :  1    -20     22      -4      1
%e A346083   5 :  1     61     90      50      5      1
%e A346083   6 :  1   -182    511    -260     95     -6     1
%e A346083   7 :  1    547   2373    2331    595    161     7    1
%e A346083   8 :  1  -1640  12412  -13944   7686  -1176   252   -8  1
%e A346083   9 :  1   4921  60420  110020  55230  20622  2100  372  9  1
%e A346083   etc.
%o A346083 (Python)
%o A346083 from functools import cache
%o A346083 @cache
%o A346083 def T(n, k):
%o A346083     if k == 0 or k == n: return 1
%o A346083     return T(n-1, k-1) + (-1)**k*(2*k + 1)*T(n-1, k)
%o A346083 for n in range(10):
%o A346083     print([T(n, k) for k in range(n+1)]) # _Peter Luschny_, Jul 22 2021
%Y A346083 Cf. A000012 (column 0 and main diagonal), A014983 (column 1), A181983 (1st subdiagonal), A002412 (2nd subdiagonal), A264851 (3rd subdiagonal without signs).
%Y A346083 Cf. A051159.
%K A346083 sign,easy,tabl
%O A346083 0,5
%A A346083 _Werner Schulte_, Jul 04 2021