OFFSET
0,2
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
T(n, k) = t(n-k, k), where 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) = 7*x + 3.
Sum_{k=0..n} T(n, k) = A049209(n).
From G. C. Greubel, Feb 22 2022: (Start)
t(k, n) = t(n, k).
T(n, n-k) = T(n, k).
t(0, n) = T(n, 0) = A000244(n). (End)
EXAMPLE
Array t(n, k) begins as:
1, 3, 9, 27, 81, ... A000244;
3, 60, 753, 8178, 84291, ...;
9, 753, 25602, 631506, 13348623, ...;
27, 8178, 631506, 30312288, 1141302225, ...;
81, 84291, 13348623, 1141302225, 70760737950, ...;
243, 852144, 259308063, 37244959794, 3608891348622, ...;
729, 8554245, 4793178096, 1109572049376, 161806374029202, ...;
Triangle, T(n, k) begins as:
1;
3, 3;
9, 60, 9;
27, 753, 753, 27;
81, 8178, 25602, 8178, 81;
243, 84291, 631506, 631506, 84291, 243;
729, 852144, 13348623, 30312288, 13348623, 852144, 729;
2187, 8554245, 259308063, 1141302225, 1141302225, 259308063, 8554245, 2187;
MATHEMATICA
f[n_]:= 7*n+3;
t[n_, k_]:= t[n, k]= If[n<0 || k<0, 0, If[n==0 && k==0, 1, f[k]*t[n-1, k] +f[n]*t[n, k-1]]];
T[n_, k_]= t[n-k, k];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 22 2022 *)
PROG
(Sage)
def f(n): return 7*n+3
@CachedFunction
def t(n, k):
if (n<0 or k<0): return 0
elif (n==0 and k==0): return 1
else: return f(k)*t(n-1, k) + f(n)*t(n, k-1)
def A257627(n, k): return t(n-k, k)
flatten([[A257627(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 22 2022
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Dale Gerdemann, May 10 2015
STATUS
approved