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A257618
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) = 8*x + 2.
9
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
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) = 8*x + 2.
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)
EXAMPLE
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;
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, 8, 2], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 24 2022 *)
PROG
(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
CROSSREFS
Cf. A000079, A144828 (row sums), A167884.
Similar sequences listed in A256890.
Sequence in context: A300361 A257617 A309344 * A088895 A257619 A075806
KEYWORD
nonn,tabl
AUTHOR
Dale Gerdemann, May 09 2015
STATUS
approved