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

0,5

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

Cf. A003462, A014915, A081271, A119258.

Sequence in context: A209414 A193636 A232968 * A144447 A051455 A346875

Adjacent sequences: A119670 A119671 A119672 * A119674 A119675 A119676

Keyword

easy,nonn,tabl

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