STATUS
reviewed
approved
The OEIS is supported by the many generous donors to the OEIS Foundation.
reviewed
approved
proposed
reviewed
editing
proposed
Row sums are: 1, 4, 48, 960, 26880, 967680, 42577920, 2214051840, 132843110400, 9033331507200, ... (see A144828).
G. C. Greubel, <a href="/A257618/b257618.txt">Rows n = 0..50 of the triangle, flattened</a>
Sum_{k=0..n} T(n, k) = A144828(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 = 8, and b = 2.
T(n, n-k) = T(n, k).
T(n, 0) = A000079(n).
T(n, 1) = 2^(n-1)*(5^n - 2*n - 1).
T(n, 2) = 2^(n-3)*(3^(2*n+1) -2*(2*n+1)*5^n -1 +4*n^2). (End)
Triangle begins as:
1;
2 , 2;
4 , 40 , 4;
8 , 472 , 472 , 8;
16 , 4928 , 16992 , 4928 , 16;
32 , 49824 , 433984 , 433984 , 49824 , 32;
64 , 499584 , 9505728 , 22567168 , 9505728 , 499584 , 64;
128 , 4999040 , 192085632 , 909941120 , 909941120 , 192085632 , 4999040 , 128;
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, 8, 2], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 24 2022 *)
(Sage)
def T(n, k, a, b): # A257618
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, 8, 2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 24 2022
approved
editing
editing
approved
editing
approved
editing
approved