OFFSET
0,2
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
T(n,k) = t(n-k, k); t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 6.
Sum_{k=0..n} T(n, k) = A051609(n).
T(n, k) = (a*k + b)*T(n-1, k) + (a*(n-k) + b)*T(n-1, k-1), with T(n, 0) = 1, a = 3, and b = 6. - G. C. Greubel, Mar 20 2022
EXAMPLE
Triangle begins as:
1;
6, 6;
36, 108, 36;
216, 1404, 1404, 216;
1296, 15876, 33696, 15876, 1296;
7776, 166212, 642492, 642492, 166212, 7776;
46656, 1659204, 10701720, 19274760, 10701720, 1659204, 46656;
279936, 16052580, 163263924, 481752360, 481752360, 163263924, 16052580, 279936;
MATHEMATICA
T[n_, k_, a_, b_]:= T[n, k, a, b]= If[k<0 || k>n, 0, If[n==0, 1, (a*(n-k)+b)*T[n-1, k-1, a, b] + (a*k+b)*T[n-1, k, a, b]]];
Table[T[n, k, 3, 6], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 20 2022 *)
PROG
(Sage)
def T(n, k, a, b): # A257626
if (k<0 or k>n): return 0
elif (n==0): return 1
else: return (a*k+b)*T(n-1, k, a, b) + (a*(n-k)+b)*T(n-1, k-1, a, b)
flatten([[T(n, k, 3, 6) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 20 2022
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Dale Gerdemann, May 10 2015
STATUS
approved