OFFSET
0,2
COMMENTS
Corresponding entries in this triangle and in A060187 differ only by powers of 2. - F. Chapoton, Nov 04 2020
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) = 4*x + 2.
Sum_{k=0..n} T(n,k) = A047053(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 = 4, and b = 2. - G. C. Greubel, Mar 20 2022
EXAMPLE
Triangle begins as:
1;
2, 2;
4, 24, 4;
8, 184, 184, 8;
16, 1216, 3680, 1216, 16;
32, 7584, 53824, 53824, 7584, 32;
64, 46208, 674752, 1507072, 674752, 46208, 64;
128, 278912, 7764096, 33244544, 33244544, 7764096, 278912, 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, 4, 2], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 20 2022 *)
PROG
(PARI) f(x) = 4*x + 2;
T(n, k) = t(n-k, k);
t(n, m) = if (!n && !m, 1, if (n < 0 || m < 0, 0, f(m)*t(n-1, m) + f(n)*t(n, m-1)));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", "); ); print(); ); \\ Michel Marcus, May 06 2015
(Sage)
def T(n, k, a, b): # A257612
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, 4, 2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 20 2022
CROSSREFS
See similar sequences listed in A256890.
KEYWORD
nonn,tabl
AUTHOR
Dale Gerdemann, May 06 2015
STATUS
approved