OFFSET
1,4
LINKS
G. C. Greubel, Rows n = 1..50 of the triangle, flattened
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)
EXAMPLE
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;
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];
Table[A171274[n, k], {n, 12}, {k, n}]//Flatten (* modified by G. C. Greubel, Mar 18 2022 *)
PROG
(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
flatten([[A171274(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 18 2022
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Roger L. Bagula and Mats Granvik, Dec 06 2009
STATUS
approved