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

Author

Dale Gerdemann, May 09 2015

Keywords

• nonn
• tabl

Examples

			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;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A000244, A008292, A034001 (row sums), A038221, A142458, A257180, A257610.

Cf. A257611, A257621, A257622, A257623, A257624, A257625, A257626, A257627.

Similar sequences listed in A256890.

Programs

Magma

A257620:= func< n,k | 3^n*EulerianNumber(n+1, k) >;
[A257620(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 17 2025

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 *)

Python

from sage.all import *
from sage.combinat.combinat import eulerian_number
def A257620(n,k): return pow(3,n)*eulerian_number(n+1,k)
print(flatten([[A257620(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Jan 17 2025

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

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)

T(n, k) = 3^n*A008292(n, k). - G. C. Greubel, Jan 17 2025