OFFSET
0,2
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
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) = x + 4.
Sum_{k=0..n} T(n, k) = A049388(n).
T(n,0) = T(n,n) = 4^n. - Georg Fischer, Oct 02 2021
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 = 1, and b = 4.
T(n, n-k) = T(n, k).
T(n, 1) = 8*5^n - 4^n*(8+n).
T(n, 2) = 2*((56 +15*n +n^2)*4^(n-1) - 4*(8+n)*5^n + 3*6^(n+1)). (End)
EXAMPLE
Triangle begins as:
1;
4, 4;
16, 40, 16;
64, 296, 296, 64;
256, 1928, 3552, 1928, 256;
1024, 11688, 34808, 34808, 11688, 1024;
4096, 67656, 302352, 487312, 302352, 67656, 4096;
16384, 379240, 2423016, 5830000, 5830000, 2423016, 379240, 16384;
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, 1, 4], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 24 2022 *)
PROG
(Sage)
def T(n, k, a, b): # A257606
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, 1, 4) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 24 2022
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Dale Gerdemann, May 03 2015
EXTENSIONS
a(3) corrected by Georg Fischer, Oct 02 2021
STATUS
approved