A257615 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) = 2*x + 5.

1, 5, 5, 25, 70, 25, 125, 715, 715, 125, 625, 6380, 12870, 6380, 625, 3125, 52785, 186010, 186010, 52785, 3125, 15625, 416370, 2360295, 4092220, 2360295, 416370, 15625, 78125, 3180215, 27488205, 75698255, 75698255, 27488205, 3180215, 78125

Offset

0,2

Links

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

Formula

T(n,k) = t(n-k, k); 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 f(x) = 2*x + 5.

Sum_{k=0..n} T(n, k) = A051582(n).

From G. C. Greubel, Mar 21 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 = 2, and b = 5.

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

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

T(n, 1) = 5*7^n - 5^n*(n+5). (End)

Example

Triangle begins as:
      1;
      5,       5;
     25,      70,       25;
    125,     715,      715,      125;
    625,    6380,    12870,     6380,      625;
   3125,   52785,   186010,   186010,    52785,     3125;
  15625,  416370,  2360295,  4092220,  2360295,   416370,   15625;
  78125, 3180215, 27488205, 75698255, 75698255, 27488205, 3180215, 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, 2, 5], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 21 2022 *)

Prog

(Sage)
def T(n, k, a, b): # A257610
    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, 2, 5) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 21 2022

Crossrefs

Cf. A000351, A051582 (row sums), A060187, A257609, A257611, A257613.

Cf. A257607, A257624.

Similar sequences listed in A256890.

Sequence in context: A093643 A223263 A189318 * A257624 A176160 A222281

Adjacent sequences: A257612 A257613 A257614 * A257616 A257617 A257618

Keyword

nonn,tabl

Author

Dale Gerdemann, May 09 2015

Status

approved