A118438 Triangle T, read by rows, equal to the matrix product T = H*C*H, where H is the self-inverse triangle A118433 and C is Pascal's triangle.
1, -1, 1, 5, -2, 1, 11, -9, -3, 1, -23, 44, 30, -4, 1, -41, 125, 110, -30, -5, 1, 45, -246, -345, 220, 75, -6, 1, -29, -301, -861, 875, 385, -63, -7, 1, 337, -232, 1260, -2296, -1610, 616, 140, -8, 1, 1199, -3015, -1044, -3612, -5166, 3150, 924, -108, -9, 1
Offset: 0
Examples
Triangle begins: 1; -1, 1; 5,-2, 1; 11,-9,-3, 1; -23, 44, 30,-4, 1; -41, 125, 110,-30,-5, 1; 45,-246,-345, 220, 75,-6, 1; -29,-301,-861, 875, 385,-63,-7, 1; 337,-232, 1260,-2296,-1610, 616, 140,-8, 1; 1199,-3015,-1044,-3612,-5166, 3150, 924,-108,-9, 1; ...
Crossrefs
Programs
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Mathematica
nmax = 12; h[n_, k_] := Binomial[n, k]*(-1)^(Quotient[n+1, 2] - Quotient[k, 2]+n-k); H = Table[h[n, k], {n, 0, nmax}, {k, 0, nmax}]; Cn = Table[Binomial[n, k], {n, 0, nmax}, {k, 0, nmax}]; Tn = H.Cn.H; T[n_, k_] := Tn[[n+1, k+1]]; Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 08 2024 *) -
PARI
{T(n,k)=local(M=matrix(n+1,n+1,r,c,if(r>=c,binomial(r-1,c-1)*(-1)^(r\2- (c-1)\2+r-c))),C=matrix(n+1,n+1,r,c,if(r>=c,binomial(r-1,c-1))));(M*C*M)[n+1,k+1]}
Formula
Since T + T^-1 = C + C^-1, then [T^-1](n,k) = (1+(-1)^(n-k))*C(n,k) - T(n,k) is a formula for the matrix inverse T^-1 = A118435.
Comments