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 + 2.
Sum_{k=0..n} T(n, k) = A008546(n).
From G. C. Greubel, Mar 01 2022: (Start)
t(k, n) = t(n, k).
T(n, n-k) = T(n, k).
t(0, n) = T(n, 0) = A000079(n). (End)
EXAMPLE
Array t(n,k) begins as:
1, 2, 4, 8, 16, ... A000079;
2, 28, 244, 1844, 13260, ...;
4, 244, 5856, 101620, 1511160, ...;
8, 1844, 101620, 3455080, 91981880, ...;
16, 13260, 1511160, 91981880, 4047202720, ...;
32, 93684, 20663388, 2121603436, 146321752612, ...;
64, 657836, 269011408, 44675623468, 4648698508440, ...;
Triangle T(n,k) begins as:
1;
2, 2;
4, 28, 4;
8, 244, 244, 8;
16, 1844, 5856, 1844, 16;
32, 13260, 101620, 101620, 13260, 32;
64, 93684, 1511160, 3455080, 1511160, 93684, 64;
128, 657836, 20663388, 91981880, 91981880, 20663388, 657836, 128;
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, 2], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 01 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 A257614(n, k): return t(n-k, k, 5, 2)
flatten([[A257614(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 01 2022
CROSSREFS
Similar sequences listed in A256890.
KEYWORD
nonn,tabl
AUTHOR
Dale Gerdemann, May 09 2015
STATUS
approved