A257611 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(n) = 2*n + 3.

1, 3, 3, 9, 30, 9, 27, 213, 213, 27, 81, 1308, 2982, 1308, 81, 243, 7431, 32646, 32646, 7431, 243, 729, 40314, 310263, 587628, 310263, 40314, 729, 2187, 212505, 2695923, 8701545, 8701545, 2695923, 212505, 2187, 6561, 1099704, 22059036, 113360904, 191433990, 113360904, 22059036, 1099704, 6561

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) = 2*n + 3.

Sum_{k=0..n} T(n, k) = A051578(n).

From G. C. Greubel, Feb 28 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, ...;
    3,     30,      213,       1308,        7431,         40314, ...;
    9,    213,     2982,      32646,      310263,       2695923, ...;
   27,   1308,    32646,     587628,     8701545,     113360904, ...;
   81,   7431,   310263,    8701545,   191433990,    3579465642, ...;
  243,  40314,  2695923,  113360904,  3579465642,   93066106692, ...;
  729, 212505, 22059036, 1351133676, 59641127202, 2104476295026, ...;
Triangle T(n,k) begins as:
     1;
     3,      3;
     9,     30,       9;
    27,    213,     213,      27;
    81,   1308,    2982,    1308,      81;
   243,   7431,   32646,   32646,    7431,     243;
   729,  40314,  310263,  587628,  310263,   40314,    729;
  2187, 212505, 2695923, 8701545, 8701545, 2695923, 212505, 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, 2, 3], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 28 2022 *)

Prog

(PARI) f(x) = 2*x + 3;
T(n, k) = t(n-k, k);
t(n, m) = if (!n && !m, 1, if (n < 0 || m < 0, 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, May 06 2015
(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 A257611(n, k): return t(n-k, k, 2, 3)
flatten([[A257611(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 28 2022

Crossrefs

Cf. A000244, A051578 (row sums), A060187, A257609, A257613, A257615.

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

Similar sequences listed in A256890.

Sequence in context: A010098 A029857 A327712 * A268617 A264412 A202889

Adjacent sequences: A257608 A257609 A257610 * A257612 A257613 A257614

Keyword

nonn,tabl

Author

Dale Gerdemann, May 06 2015

Status

approved