A119673 T(n, k) = 3*T(n-1, k-1) + T(n-1, k) for k < n and T(n, n) = 1, T(n, k) = 0, if k < 0 or k > n; triangle read by rows.

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, 1, 28, 349, 2542, 11926, 37384, 78322, 105796, 83653, 29524, 1

Offset: 0

Author

Zerinvary Lajos, Jun 11 2006

Keywords

• easy
• nonn
• tabl

Examples

			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;
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened

Crossrefs

Cf. A003462, A014915, A081271, A119258.

Programs

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

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, ] = 0;
Table[T[n, k], {n, 0, 10}, {k, 0, n}] (* Jean-François Alcover, Jun 13 2019 *)

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

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

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

Extensions

Definition clarified by Philippe Deléham, Jun 13 2006

Entry revised by N. J. A. Sloane, Jun 19 2006