A257625 Triangle read by rows: 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) = 6*n + 3.

1, 3, 3, 9, 54, 9, 27, 621, 621, 27, 81, 6156, 18630, 6156, 81, 243, 57591, 408726, 408726, 57591, 243, 729, 526338, 7685847, 17166492, 7685847, 526338, 729, 2187, 4765473, 132656859, 568014201, 568014201, 132656859, 4765473, 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), and f(n) = 6*n + 3.

Sum_{k=0..n} T(n, k) = A047058(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) = A000244(n). (End)

Example

Array t(n,k) begins as:
    1,       3,          9,           27,             81, ...;
    3,      54,        621,         6156,          57591, ...;
    9,     621,      18630,       408726,        7685847, ...;
   27,    6156,     408726,     17166492,      568014201, ...;
   81,   57591,    7685847,    568014201,    30672766854, ...;
  243,  526338,  132656859,  16305974568,  1366261865802, ...;
  729, 4765473, 2175706332, 427278012876, 53552912878818, ...;
Triangle T(n,k) begins as:
     1;
     3,       3;
     9,      54,         9;
    27,     621,       621,        27;
    81,    6156,     18630,      6156,        81;
   243,   57591,    408726,    408726,     57591,       243;
   729,  526338,   7685847,  17166492,   7685847,    526338,     729;
  2187, 4765473, 132656859, 568014201, 568014201, 132656859, 4765473, 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, 6, 3], {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 A257625(n, k): return t(n-k, k, 6, 3)
flatten([[A257625(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 01 2022

Crossrefs

Cf. A047058 (row sums), A142461, A257616.

Cf. A038221, A257180, A257611, A257620, A257621, A257623, A257625, A257627.

See similar sequences listed in A256890.

Sequence in context: A257621 A190960 A257623 * A216147 A334774 A257627

Adjacent sequences: A257622 A257623 A257624 * A257626 A257627 A257628

Keyword

nonn,tabl

Author

Dale Gerdemann, May 10 2015

Status

approved