OFFSET
0,5
LINKS
G. C. Greubel, Rows n = 0..100 of triangle, flattened
FORMULA
T(n,k) = R(n,k,3) where R(n,k,m) = (1-m)^(-n+k)-m^(k+1)*Pochhammer(n-k, k+1)* hyper2F1([1,n+1],[k+2],m)/(k+1)!. - Peter Luschny, Jul 25 2014
EXAMPLE
Triangle begins:
1;
1, 1;
1, 4, 1;
1, 7, 13, 1;
1, 10, 34, 40, 1;
1, 13, 64, 142, 121, 1;
1, 16, 103, 334, 547, 364, 1;
1, 19, 151, 643, 1549, 2005, 1093, 1;
1, 22, 208, 1096, 3478, 6652, 7108, 3280, 1;
1, 25, 274, 1720, 6766, 17086, 27064, 24604, 9841, 1;
MAPLE
T := (n, k, m) -> (1-m)^(-n+k)-m^(k+1)*pochhammer(n-k, k+1)* hypergeom([1, n+1], [k+2], m)/(k+1)!; A119673 := (n, k) -> T(n, k, 3);
seq(print(seq(round(evalf(A119673(n, k))), k=0..n)), n=0..10); # Peter Luschny, Jul 25 2014
MATHEMATICA
T[_, 0]=1; T[n_, n_]=1; T[n_, k_]/; 0<k<n := T[n, k] = 3T[n-1, k-1] + T[n-1, k]; T[_, _] = 0;
Table[T[n, k], {n, 0, 10}, {k, 0, n}] (* Jean-François Alcover, Jun 13 2019 *)
PROG
(PARI) T(n, k) = if(k<0 || k>n, 0, if(k==n, 1, 3*T(n-1, k-1) +T(n-1, k)));
for(n=0, 12, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Nov 18 2019
(Magma)
function T(n, k)
if k lt 0 or k gt n then return 0;
elif k eq n then return 1;
else return 3*T(n-1, k-1) + T(n-1, k);
end if;
return T;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 18 2019
(Sage)
@CachedFunction
def T(n, k):
if (k<0 or k>n): return 0
elif (k==n): return 1
else: return 3*T(n-1, k-1) + T(n-1, k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 18 2019
CROSSREFS
KEYWORD
AUTHOR
Zerinvary Lajos, Jun 11 2006
EXTENSIONS
Definition clarified by Philippe Deléham, Jun 13 2006
Entry revised by N. J. A. Sloane, Jun 19 2006
STATUS
approved