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
KEYWORD
nonn,tabl
AUTHOR
Dale Gerdemann, May 09 2015
STATUS
approved