A257627 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) = 7*x + 3.

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

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

Cf. A000244, A038221, A049209 (row sums), A142462.

Cf. A257180, A257611, A257617, A257620, A257621, A257623, A257625.

See similar sequences listed in A256890.

Sequence in context: A257625 A216147 A334774 * A115564 A122961 A165421

Adjacent sequences: A257624 A257625 A257626 * A257628 A257629 A257630

Keyword

nonn,tabl

Author

Dale Gerdemann, May 10 2015

Status

approved