A257620 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) = 3*n + 3.

1, 3, 3, 9, 36, 9, 27, 297, 297, 27, 81, 2106, 5346, 2106, 81, 243, 13851, 73386, 73386, 13851, 243, 729, 87480, 868239, 1761264, 868239, 87480, 729, 2187, 540189, 9388791, 34158753, 34158753, 9388791, 540189, 2187, 6561, 3293622, 95843088, 578903274, 1024762590, 578903274, 95843088, 3293622, 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) = 3*n + 3.

Sum_{k=0..n} T(n, k) = A034001(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,     36,      297,       2106,        13851,          87480, ...;
    9,    297,     5346,      73386,       868239,        9388791, ...;
   27,   2106,    73386,    1761264,     34158753,      578903274, ...;
   81,  13851,   868239,   34158753,   1024762590,    25791697782, ...;
  243,  87480,  9388791,  578903274,  25791697782,   928501120152, ...;
  729, 540189, 95843088, 8959544136, 575025893586, 28788563928042, ...;
Triangle T(n,k) begins as:
     1;
     3,      3;
     9,     36,       9;
    27,    297,     297,       27;
    81,   2106,    5346,     2106,       81;
   243,  13851,   73386,    73386,    13851,     243;
   729,  87480,  868239,  1761264,   868239,   87480,    729;
  2187, 540189, 9388791, 34158753, 34158753, 9388791, 540189, 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, 3, 3], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 28 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 A257620(n, k): return t(n-k, k, 3, 3)
flatten([[A257620(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 28 2022

Crossrefs

Cf. A000244, A034001 (row sums), A142458, A257610, A257622, A257624, A257626.

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

Similar sequences listed in A256890.

Sequence in context: A268617 A264412 A202889 * A032086 A241278 A100239

Adjacent sequences: A257617 A257618 A257619 * A257621 A257622 A257623

Keyword

nonn,tabl

Author

Dale Gerdemann, May 09 2015

Status

approved