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%I #39 Feb 12 2023 10:52:08
%S 1,-1,1,0,-1,1,0,0,-1,1,1,0,0,-2,1,-1,1,0,0,-1,1,2,-3,3,0,0,-3,1,-2,2,
%T -3,3,0,0,-1,1,9,-8,8,-12,6,0,0,-4,1,-9,9,-8,8,-6,6,0,0,-1,1,44,-45,
%U 45,-40,20,-30,10,0,0,-5,1,-44,44,-45,45,-20,20,-10,10,0,0,-1,1
%N Matrix inverse of lower triangular array A211226.
%C This triangle is related to the derangement numbers. The subtriangles (T(2*n,2*k))n,k>=0, -(T(2*n+1,2*k))n,k>=0, and (T(2*n+1,2*k+1))n,k>=0 are all equal to A008290, while the subtriangle (T(2*n,2*k+1))n,k>=0 equals -A180188 (with an extra initial row of zeros).
%F T(2*n,2*k) = T(2*n+1,2*k+1) = -T(2*n+1,2*k) = binomial(n,k)*A000166(n-k) = (n!/k!)*Sum_{i = 0..n-k} (-1)^i/i!;
%F T(2*n,2*k+1) = -n*binomial(n-1,k)*A000166(n-k-1) = -(n!/k!)*Sum_{i = 0..n-k-1} (-1)^i/i!.
%F T(n,k) = T(n-k,0)*A211226(n,k).
%F Column entries:
%F T(2*n,0) = A000166(n), T(2*n,2) = A000240(n), T(2*n,4) = A000387(n), T(2*n,6) = A000449(n), T(2*n,8) = A000475(n).
%F From _Manfred Boergens_, Jan 10 2023: (Start)
%F With b(j) = floor(j/2); h = 1 for n even and k odd, h = 0 else:
%F T(n,k) = (-1)^(n+k)*(b(n)!/b(k)!)*Sum_{i = 0..b(n-k)-h} (-1)^i/i!.
%F Sum-free formula:
%F T(n,k) = (-1)^(n+k)*(b(n)!/b(k)!) for n-k < 2.
%F T(n,k) = (-1)^(n+k)*(b(n)!/b(k)!)*round((b(n-k)-h)!/exp(1))/(b(n-k)-h)!) otherwise. (End)
%e Triangle begins:
%e n\k | 0 1 2 3 4 5 6 7 8 9
%e =====+==================================================
%e 0 | 1
%e 1 | -1 1
%e 2 | 0 -1 1
%e 3 | 0 0 -1 1
%e 4 | 1 0 0 -2 1
%e 5 | -1 1 0 0 -1 1
%e 6 | 2 -3 3 0 0 -3 1
%e 7 | -2 2 -3 3 0 0 -1 1
%e 8 | 9 -8 8 -12 6 0 0 -4 1
%e 9 | -9 9 -8 8 -6 6 0 0 -1 1
%e ...
%t b[j_] = Floor[j/2]; h = If[EvenQ[n] && OddQ[k], 1, 0];
%t Table[(-1)^(n+k) (b[n]!/b[k]!) Sum[(-1)^i/i!, {i, 0, b[n-k]-h}], {n, 0, 31}, {k, 0, n}] //Flatten (* _Manfred Boergens_, Jan 10 2023 *)
%t (* Sum-free code *)
%t b[j_] = Floor[j/2]; h = If[EvenQ[n] && OddQ[k], 1, 0];
%t T[n_, k_] = (-1)^(n+k) (b[n]!/b[k]!) If[n-k<2, 1, Round[(b[n-k]-h)!/E]/(b[n-k]-h)!];
%t Table[T[n, k], {n, 0, 31}, {k, 0, n}] // Flatten
%t (* _Manfred Boergens_, Jan 10 2023 *)
%o (PARI) f(n) = (n\2)!; \\ A081123
%o T(n,k) = f(n)/(f(k)*f(n-k)); \\ A211226
%o tabl(nn) = my(m=matrix(nn, nn, n, k, if (n>=k, T(n-1,k-1), 0))); 1/m; \\ _Michel Marcus_, Jan 10 2023
%Y Cf. A000166, A000240, A000387, A000449, A000475, A008290, A180188, A211226.
%K sign,easy,tabl
%O 0,14
%A _Peter Bala_, Apr 05 2012
%E More terms from _Manfred Boergens_, Jan 10 2023
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