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), and f(n) = 5*n + 3.
Sum_{k=0..n} T(n, k) = A008548(n).
From G. C. Greubel, Feb 27 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, 48, 501, 4494, 37815, ...;
9, 501, 13026, 250230, 4122735, ...;
27, 4494, 250230, 9008280, 256971945, ...;
81, 37815, 4122735, 256971945, 11820709470, ...;
243, 309324, 62256627, 6368680566, 450199373658, ...;
729, 2498649, 891791568, 144065371932, 15108742867890, ...;
Triangle, T(n,k), begins as:
1;
3, 3;
9, 48, 9;
27, 501, 501, 27;
81, 4494, 13026, 4494, 81;
243, 37815, 250230, 250230, 37815, 243;
729, 309324, 4122735, 9008280, 4122735, 309324, 729;
2187, 2498649, 62256627, 256971945, 256971945, 62256627, 2498649, 2187;
MATHEMATICA
t[n_, k_, p_, q_]:= t[n, k, p, q]= If[n<0 || k<0, 0, If[n==0 && k==0, 1, (p*k+ q)*t[n-1, k, p, q] + (p*n+q)*t[n, k-1, p, q]]];
T[n_, k_, p_, q_]= t[n-k, k, p, q];
Table[T[n, k, 5, 3], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 27 2022 *)
PROG
(Sage)
@CachedFunction
def t(n, k, p, q):
if (n<0 or k<0): return 0
elif (n==0 and k==0): return 1
else: return (p*k+q)*t(n-1, k, p, q) + (p*n+q)*t(n, k-1, p, q)
def A257623(n, k): return t(n-k, k, 5, 3)
flatten([[A257623(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 27 2022
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Dale Gerdemann, May 10 2015
STATUS
approved