A257621 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) = 4*n + 3.

1, 3, 3, 9, 42, 9, 27, 393, 393, 27, 81, 3156, 8646, 3156, 81, 243, 23631, 142446, 142446, 23631, 243, 729, 171006, 2015895, 4273380, 2015895, 171006, 729, 2187, 1216725, 26107983, 102402705, 102402705, 26107983, 1216725, 2187, 6561, 8584872, 320039388, 2136524184, 3891302790, 2136524184, 320039388, 8584872, 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) = 4*n + 3.

Sum_{k=0..n} T(n, k) = A000407(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,      42,       393,        3156,         23631, ...;
    9,     393,      8646,      142446,       2015895, ...;
   27,    3156,    142446,     4273380,     102402705, ...;
   81,   23631,   2015895,   102402705,    3891302790, ...;
  243,  171006,  26107983,  2136524184,  123074809242, ...;
  729, 1216725, 320039388, 40688926236, 3437022383970, ...;
Triangle T(n,k) begins as:
     1;
     3,       3;
     9,      42,        9;
    27,     393,      393,        27;
    81,    3156,     8646,      3156,        81;
   243,   23631,   142446,    142446,     23631,      243;
   729,  171006,  2015895,   4273380,   2015895,   171006,     729;
  2187, 1216725, 26107983, 102402705, 102402705, 26107983, 1216725, 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, 4, 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 A257621(n, k): return t(n-k, k, 4, 3)
flatten([[A257621(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 01 2022

Crossrefs

Cf. A000407 (row sums), A142459, A257612.

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

Similar sequences listed in A256890.

Sequence in context: A100239 A245023 A038080 * A190960 A257623 A257625

Adjacent sequences: A257618 A257619 A257620 * A257622 A257623 A257624

Keyword

nonn,tabl

Author

Dale Gerdemann, May 09 2015

Status

approved