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A378931 Triangle read by rows, based on products of Jacobsthal numbers (A001045).

%I A378931 #6 Dec 21 2024 01:01:43
%S A378931 1,-1,3,-2,-9,15,-4,-18,-25,55,-8,-36,-50,-121,231,-16,-72,-100,-242,
%T A378931 -441,903,-32,-144,-200,-484,-882,-1849,3655,-64,-288,-400,-968,-1764,
%U A378931 -3698,-7225,14535,-128,-576,-800,-1936,-3528,-7396,-14450,-29241,58311,-256,-1152,-1600,-3872,-7056,-14792,-28900,-58482,-116281,232903
%N A378931 Triangle read by rows, based on products of Jacobsthal numbers (A001045).
%C A378931 Let M = T^(-1) be matrix inverse of T seen as a lower triangular matrix. M is a harmonic triangle with M(n, k) = 1 / A084175(n) if k = n, and 1 / A378676(k) if 1 <= k < n. Triangle M(n, k) for 1 <= k <= n starts:
%C A378931   1/1
%C A378931   1/3  1/3
%C A378931   1/3  1/5  1/15
%C A378931   1/3  1/5  1/33   1/55
%C A378931   1/3  1/5  1/33  1/105  1/231
%C A378931   1/3  1/5  1/33  1/105  1/473  1/903
%C A378931   etc.
%C A378931   Sum_{k=1..n} M(n, k) * 2^(k-1) = 1.
%C A378931   Sum_{k=1..n} M(n, k) * (-2)^(k-1) = (-1)^(n-1) / A001045(n+1).
%C A378931   Sum_{k=1..n} 2^(k-1) / M(n, k) = (8^n - 1) / 7 = A023001(n).
%F A378931 T(n, n) = (2 * 4^n - (-2)^n - 1) / 9 = A084175(n), and T(n, k) = -2^(n-1-k) * (2^(k+1) + (-1)^k)^2 / 9 for 1 <= k < n.
%F A378931 G.f.: x*t * (1 - 3*t - 6*x*t^2 + 8*x^2*t^3) / ((1 - 2*t) * (1 - x*t) * (1 + 2*x*t) * (1 - 4*x*t)).
%e A378931 Triangle T(n, k) for 1 <= k <= n starts:
%e A378931 n\k :     1     2     3      4      5      6       7       8      9
%e A378931 ===================================================================
%e A378931   1 :     1
%e A378931   2 :    -1     3
%e A378931   3 :    -2    -9    15
%e A378931   4 :    -4   -18   -25     55
%e A378931   5 :    -8   -36   -50   -121    231
%e A378931   6 :   -16   -72  -100   -242   -441    903
%e A378931   7 :   -32  -144  -200   -484   -882  -1849    3655
%e A378931   8 :   -64  -288  -400   -968  -1764  -3698   -7225   14535
%e A378931   9 :  -128  -576  -800  -1936  -3528  -7396  -14450  -29241  58311
%e A378931   etc.
%t A378931 T[n_,k_]:=If[k==n, (2*4^n-(-2)^n-1)/9, -2^(n-1-k)*(2^(k+1)+(-1)^k)^2/9]; Table[T[n,k],{n,10},{k,n}]//Flatten (* _Stefano Spezia_, Dec 11 2024 *)
%o A378931 (PARI) T(n,k)=if(k==n,(2*4^n-(-2)^n-1)/9,-2^(n-1-k)*(2^(k+1)+(-1)^k)^2/9)
%Y A378931 Cf. A001045, A023001, A378676.
%Y A378931 A084175 (main diagonal), A139818 (1st subdiagonal), A000079 (column 1 and row sums).
%K A378931 sign,easy,tabl
%O A378931 1,3
%A A378931 _Werner Schulte_, Dec 11 2024