OFFSET
0,2
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) = 2*x + 2.
Sum_{k=0..n} T(n, k) = A002866(n).
From G. C. Greubel, Mar 21 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 = 2, and b = 2.
T(n, n-k) = T(n, k).
T(n, 0) = A000079(n).
T(n, 1) = 2*A100575(n+1). (End)
EXAMPLE
Triangle begins as:
1;
2, 2;
4, 16, 4;
8, 88, 88, 8;
16, 416, 1056, 416, 16;
32, 1824, 9664, 9664, 1824, 32;
64, 7680, 76224, 154624, 76224, 7680, 64;
128, 31616, 549504, 1999232, 1999232, 549504, 31616, 128;
256, 128512, 3739648, 22587904, 39984640, 22587904, 3739648, 128512, 256;
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, 2, 2], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 21 2022 *)
PROG
(Magma)
function T(n, k, a, b)
if k lt 0 or k gt n then return 0;
elif k eq 0 or k eq n then return 1;
else return (a*k+b)*T(n-1, k, a, b) + (a*(n-k)+b)*T(n-1, k-1, a, b);
end if; return T;
end function;
[T(n, k, 2, 2): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 21 2022
(Sage)
def T(n, k, a, b): # A257609
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, 2, 2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 21 2022
CROSSREFS
Cf. similar sequences listed in A256890.
KEYWORD
nonn,tabl
AUTHOR
Dale Gerdemann, May 03 2015
STATUS
approved