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A257624
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) = 3*x + 5.
8
1, 5, 5, 25, 80, 25, 125, 915, 915, 125, 625, 9070, 20130, 9070, 625, 3125, 83185, 348410, 348410, 83185, 3125, 15625, 727980, 5246655, 9755480, 5246655, 727980, 15625, 78125, 6183215, 72272805, 225769855, 225769855, 72272805, 6183215, 78125
OFFSET
0,2
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) = 3*x + 5.
Sum_{k=0..n} T(n, k) = A051607(n).
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 = 3, and b = 5. - G. C. Greubel, Mar 20 2022
EXAMPLE
Triangle begins as:
1;
5, 5;
25, 80, 25;
125, 915, 915, 125;
625, 9070, 20130, 9070, 625;
3125, 83185, 348410, 348410, 83185, 3125;
15625, 727980, 5246655, 9755480, 5246655, 727980, 15625;
78125, 6183215, 72272805, 225769855, 225769855, 72272805, 6183215, 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, 3, 5], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 20 2022 *)
PROG
(Sage)
def T(n, k, a, b): # A257624
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, 3, 5) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 20 2022
CROSSREFS
Similar sequences listed in A256890.
Sequence in context: A223263 A189318 A257615 * A176160 A222281 A214706
KEYWORD
nonn,tabl
AUTHOR
Dale Gerdemann, May 10 2015
STATUS
approved