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(x) = x + 3.
Sum_{k=0..n} T(n, k) = A001725(n+5).
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, 243, ... A000244;
3, 24, 141, 726, 3471, 15828, ...;
9, 141, 1410, 11406, 81327, 533259, ...;
27, 726, 11406, 136872, 1390521, 12609198, ...;
81, 3471, 81327, 1390521, 19467294, 237144642, ...;
243, 15828, 533259, 12609198, 237144642, 3794314272, ...;
729, 69873, 3295152, 105311556, 2607816498, 53824862658, ...;
Triangle T(n,k) begins as:
1;
3, 3;
9, 24, 9;
27, 141, 141, 27;
81, 726, 1410, 726, 81;
243, 3471, 11406, 11406, 3471, 243;
729, 15828, 81327, 136872, 81327, 15828, 729;
2187, 69873, 533259, 1390521, 1390521, 533259, 69873, 2187;
6561, 301362, 3295152, 12609198, 19467294, 12609198, 3295152, 301362, 6561;
MATHEMATICA
f[n_]:= 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
(PARI) f(x) = x + 3;
T(n, k) = t(n-k, k);
t(n, m) = {if (!n && !m, return(1)); if (n < 0 || m < 0, return (0)); f(m)*t(n-1, m) + f(n)*t(n, m-1); }
tabl(nn) = {for (n=0, nn, for (k=0, n, print1(T(n, k), ", "); ); print(); ); } \\ Michel Marcus, Apr 23 2015
(Sage)
def f(n): return 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
Similar sequences listed in A256890.
KEYWORD
nonn,tabl
AUTHOR
Dale Gerdemann, Apr 17 2015
STATUS
approved