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%I #18 Mar 18 2022 07:00:40
%S 1,-1,1,7,-8,1,-235,273,-39,1,35353,-41116,5928,-166,1,-22683409,
%T 26382125,-3804940,106900,-677,1,60147266239,-69954818244,10089231945,
%U -283474190,1796973,-2724,1,-648088191536203,753764796604717,-108711714513099,3054442698125,-19362601277,29358651,-10915,1
%N Matrix inverse of A142458.
%H G. C. Greubel, <a href="/A171274/b171274.txt">Rows n = 1..50 of the triangle, flattened</a>
%F Sum_{j=k..n} T(n,j)*A142458(j,k) = delta(n,k), the Kronecker delta.
%F 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
%F From _G. C. Greubel_, Mar 18 2022: (Start)
%F Sum_{k=1..n} T(n, k) = 0^(n-1).
%F T(n, n-1) = (-1)*A142458(n, 2). (End)
%e The triangle starts as:
%e 1;
%e -1, 1;
%e 7, -8, 1;
%e -235, 273, -39, 1;
%e 35353, -41116, 5928, -166, 1;
%e -22683409, 26382125, -3804940, 106900, -677, 1;
%e 60147266239, -69954818244, 10089231945, -283474190, 1796973, -2724, 1;
%p 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:
%p 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:
%p seq(seq(A171274(n,k), k=1..n), n=1..10); # _R. J. Mathar_, Jun 04 2011
%t 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]];
%t A142458[n_, k_]:= T[n,k,3];
%t A171274[n_, k_]:= A171274[n, k]= If[k==n, 1, -Sum[A171274[n, j]*A142458[j, k], {j,k+1,n}] ];
%t Table[A171274[n, k], {n,12}, {k,n}]//Flatten (* modified by _G. C. Greubel_, Mar 18 2022 *)
%o (Sage)
%o def T(n,k,m):
%o if (k==1 or k==n): return 1
%o else: return (m*(n-k)+1)*T(n-1,k-1,m) + (m*k-m+1)*T(n-1,k,m)
%o def A142458(n,k): return T(n,k,3)
%o @CachedFunction
%o def A171274(n,k):
%o if (k==n): return 1
%o else: return (-1)*sum( A171274(n,j)*A142458(j,k) for j in (k+1..n) )
%o flatten([[A171274(n,k) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Mar 18 2022
%Y Cf. A142458.
%K sign,tabl
%O 1,4
%A _Roger L. Bagula_ and _Mats Granvik_, Dec 06 2009
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