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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), where and f(xn) = 6*x n + 3.

Row sums are: 1, 6, 72, 1296, 31104, 933120, 33592320, 1410877440, 67722117120, 3656994324480, ... (see A047058).

G. C. Greubel, <a href="/A257625/b257625.txt">Rows n = 0..50 of the triangle, flattened</a>

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 and f(xn) = 6*x 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)

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;

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 *)

(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

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

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Cf. A142461, A257616, A257633.

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Cf. A142461, A257616, A257633, A257640, A257648, A257650, A257652.

Cf. A038221, A257180, A257611, A257620, A257621, A257623, A257625, A257627, A257628, A257629.

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