A171274 Matrix inverse of A142458.

1, -1, 1, 7, -8, 1, -235, 273, -39, 1, 35353, -41116, 5928, -166, 1, -22683409, 26382125, -3804940, 106900, -677, 1, 60147266239, -69954818244, 10089231945, -283474190, 1796973, -2724, 1, -648088191536203, 753764796604717, -108711714513099, 3054442698125, -19362601277, 29358651, -10915, 1

Offset: 1

Author

Roger L. Bagula and Mats Granvik, Dec 06 2009

Keywords

• sign
• tabl

Examples

			The triangle starts as:
            1;
           -1,            1;
            7,           -8,           1;
         -235,          273,         -39,          1;
        35353,       -41116,        5928,       -166,       1;
    -22683409,     26382125,    -3804940,     106900,    -677,     1;
  60147266239, -69954818244, 10089231945, -283474190, 1796973, -2724, 1;
		

Links

• G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Crossrefs

Cf. A142458.

Programs

Maple

A142458:= proc(n,k) if n = k then 1; elif k > n or k < 1 then 0 ; else (3*n-3*k+1)*procname(n-1,k-1)+(3*k-2)*procname(n-1,k) ; end if; end proc:
A171274 := proc(n,k) option remember; if k=n then 1; else -add( procname(n,j)*A142458(j,k),j=k+1..n);  end if; end proc:
seq(seq(A171274(n,k), k=1..n), n=1..10); # R. J. Mathar, Jun 04 2011

Mathematica

T[n_, k_, m_]:= T[n, k, m]= If[k==1 || k==n, 1, (m*n-m*k+1)*T[n-1, k-1, m] + (m*k -m+1)*T[n-1, k, m]];
A142458[n_, k_]:= T[n,k,3];
A171274[n_, k_]:= A171274[n, k]= If[k==n, 1, -Sum[A171274[n, j]*A142458[j, k], {j,k+1,n}] ];
Table[A171274[n, k], {n,12}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 18 2022 *)

Sage

def T(n,k,m):
    if (k==1 or k==n): return 1
    else: return (m*(n-k)+1)*T(n-1,k-1,m) + (m*k-m+1)*T(n-1,k,m)
def A142458(n,k): return T(n,k,3)
@CachedFunction
def A171274(n,k):
    if (k==n): return 1
    else: return (-1)*sum( A171274(n,j)*A142458(j,k) for j in (k+1..n) )
flatten([[A171274(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 18 2022

Formula

Sum_{j=k..n} T(n,j)*A142458(j,k) = delta(n,k), the Kronecker delta.

T(n, k) = (-1)*Sum_{j=k+1..n} T(n, j)*A142458(j, k), with T(n, n) = 1. - R. J. Mathar, Jun 04 2011

From G. C. Greubel, Mar 18 2022: (Start)

Sum_{k=1..n} T(n, k) = 0^(n-1).

T(n, n-1) = (-1)*A142458(n, 2). (End)