A257614 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) = 5*n + 2.

1, 2, 2, 4, 28, 4, 8, 244, 244, 8, 16, 1844, 5856, 1844, 16, 32, 13260, 101620, 101620, 13260, 32, 64, 93684, 1511160, 3455080, 1511160, 93684, 64, 128, 657836, 20663388, 91981880, 91981880, 20663388, 657836, 128, 256, 4609588, 269011408, 2121603436, 4047202720, 2121603436, 269011408, 4609588, 256

Offset: 0

Author

Dale Gerdemann, May 09 2015

Keywords

• nonn
• tabl

Examples

			Array t(n,k) begins as:
   1,      2,         4,           8,            16, ... A000079;
   2,     28,       244,        1844,         13260, ...;
   4,    244,      5856,      101620,       1511160, ...;
   8,   1844,    101620,     3455080,      91981880, ...;
  16,  13260,   1511160,    91981880,    4047202720, ...;
  32,  93684,  20663388,  2121603436,  146321752612, ...;
  64, 657836, 269011408, 44675623468, 4648698508440, ...;
Triangle T(n,k) begins as:
    1;
    2,      2;
    4,     28,        4;
    8,    244,      244,        8;
   16,   1844,     5856,     1844,       16;
   32,  13260,   101620,   101620,    13260,       32;
   64,  93684,  1511160,  3455080,  1511160,    93684,     64;
  128, 657836, 20663388, 91981880, 91981880, 20663388, 657836, 128;
		

Links

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

Crossrefs

Cf. A000079, A008546 (row sums), A142460, A257623.

Cf. A038208, A256890, A257609, A257610, A257612, A257616, A257617, A257618, A257619.

Similar sequences listed in A256890.

Programs

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,5,2], {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 A257614(n,k): return t(n-k,k,5,2)
flatten([[A257614(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 01 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) = 5*n + 2.

Sum_{k=0..n} T(n, k) = A008546(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) = A000079(n). (End)