OFFSET
1,3
COMMENTS
Row sums are {1, -1, -6, -27, -138, -831, -5820, -46563, -419070, -4190703, ...}.
The Mathematica code for e(n,k,m) gives eleven sequences of which the first few are in the OEIS (see Crossrefs section).
REFERENCES
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 214-215.
LINKS
G. C. Greubel, Rows n = 1..100 of triangle, flattened
FORMULA
T(n, k) = (k+3)*T(n-1, k) + (n-k-2)*T(n-1, k-1) with T(n,1) = 1, T(n,n) = (-2)^(n-1).
e(n,k,m)= (k+m)*e(n-1, k, m) + (n-k+1-m)*e(n-1, k-1, m) with m=3.
EXAMPLE
Triangle begins with:
1.
1, -2.
1, -11, 4.
1, -55, 35, -8.
1, -274, 210, -91, 16.
1, -1368, 986, -637, 219, -32.
1, -6837, 3180, -3473, 1752, -507, 64.
1, -34181, -1431, -17951, 10543, -4563, 1147, -128.
1, -170900, -145310, -129950, 48442, -30524, 11470, -2555, 256.
MATHEMATICA
e[n_, 0, m_]:= 1; (* Example for m=3 *)
e[n_, k_, m_]:= 0 /; k >= n;
e[n_, k_, m_]:= (k+m)*e[n-1, k, m] + (n-k+1-m)*e[n-1, k-1, m];
Table[Flatten[Table[Table[e[n, k, m], {k, 0, n-1}], {n, 1, 10}]], {m, 0, 10}]
T[n_, 1]:= 1; T[n_, n_]:= (-2)^(n-1); T[n_, k_]:= T[n, k] = (k+3)*T[n-1, k] + (n-k-2)*T[n-1, k-1]; Table[T[n, k], {n, 1, 12}, {k, 1, n}]//Flatten
PROG
(PARI) {T(n, k) = if(k==1, 1, if(k==n, (-2)^(n-1), (k+3)*T(n-1, k) + (n-k-2)* T(n-1, k-1)))};
for(n=1, 12, for(k=1, n, print1(T(n, k), ", ")))
(Sage)
def T(n, k):
if (k==1): return 1
elif (k==n): return (-2)^(n-1)
else: return (k+3)*T(n-1, k) + (n-k-2)* T(n-1, k-1)
[[T(n, k) for k in (1..n)] for n in (1..12)]
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
G. C. Greubel, Feb 22 2019
STATUS
approved