A257180 Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(x) = x + 3.

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, 19683, 1277619, 19489380, 105311556, 237144642, 237144642, 105311556, 19489380, 1277619, 19683

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

Cf. A000244, A008292, A001725 (row sums), A038221, A256890, A257606.

Cf. A257607, A257611, A257620, A257621, A257623, A257625, A257627.

Similar sequences listed in A256890.

Sequence in context: A007683 A316220 A059728 * A184694 A215885 A176158

Adjacent sequences: A257177 A257178 A257179 * A257181 A257182 A257183

Keyword

nonn,tabl

Author

Dale Gerdemann, Apr 17 2015

Status

approved