A257607 Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 5.

1, 5, 5, 25, 60, 25, 125, 535, 535, 125, 625, 4210, 7490, 4210, 625, 3125, 30885, 86110, 86110, 30885, 3125, 15625, 216560, 880735, 1377760, 880735, 216560, 15625, 78125, 1471235, 8330745, 18948695, 18948695, 8330745, 1471235, 78125, 390625, 9764910, 74498800, 234897010, 341076510, 234897010, 74498800, 9764910, 390625

Offset

0,2

Links

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

Formula

T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 5.

Sum_{k=0..n} T(n, k) = A049198(n).

From G. C. Greubel, Mar 24 2022: (Start)

T(n, k) = (a*k + b)*T(n-1, k) + (a*(n-k) + b)*T(n-1, k-1), with T(n, 0) = 1, a = 1, and b = 5.

T(n, n-k) = T(n, k).

T(n, 0) = A000351(n).

T(n, 1) = 10*6^n - 5^n*(10 + n).

T(n, 2) = 55*7^n - 10*6^n*(n+10) + 5^n*binomial(n+10, 2). (End)

Example

Triangle begins as:
      1;
      5,       5;
     25,      60,      25;
    125,     535,     535,      125;
    625,    4210,    7490,     4210,      625;
   3125,   30885,   86110,    86110,    30885,    3125;
  15625,  216560,  880735,  1377760,   880735,  216560,   15625;
  78125, 1471235, 8330745, 18948695, 18948695, 8330745, 1471235, 78125;

Mathematica

T[n_, k_, a_, b_]:= T[n, k, a, b]= If[k<0 || k>n, 0, If[n==0, 1, (a*(n-k)+b)*T[n-1, k-1, a, b] + (a*k+b)*T[n-1, k, a, b]]];
Table[T[n, k, 1, 5], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 24 2022 *)

Prog

(Sage)
def T(n, k, a, b): # A257607
    if (k<0 or k>n): return 0
    elif (n==0): return 1
    else: return  (a*k+b)*T(n-1, k, a, b) + (a*(n-k)+b)*T(n-1, k-1, a, b)
flatten([[T(n, k, 1, 5) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 24 2022

Crossrefs

Cf. A000351, A008292, A049198 (row sums), A256890, A257180, A257606

Cf. A257615, A257624

Similar sequences listed in A256890.

Sequence in context: A237654 A038247 A263798 * A093643 A223263 A189318

Adjacent sequences: A257604 A257605 A257606 * A257608 A257609 A257610

Keyword

nonn,tabl

Author

Dale Gerdemann, May 03 2015

Status

approved