A122960 Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, -1, -1, 1, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

1, 0, 1, 0, 1, 1, 0, 0, 3, 1, 0, 1, 0, 6, 1, 0, 0, 5, 0, 10, 1, 0, 1, 0, 15, 0, 15, 1, 0, 0, 7, 0, 35, 0, 21, 1, 0, 1, 0, 28, 0, 70, 0, 28, 1, 0, 0, 9, 0, 84, 0, 126, 0, 36, 1, 0, 1, 0, 45, 0, 210, 0, 210, 0, 45, 1

Offset

0,9

Comments

T(n,k) = binomial (n,n-k+1) if (n-k) is an odd number (see A000217, A000332, A000579, A000581, ...). T(n,k)= 0 if (n-k)=2x with x > 0 (see A000004). T(n,n)=1 (see A000012).

Links

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

Formula

Sum_{k=0..n} T(n,k) = A011782(n).

Sum_{k=0..n} 2^k*T(n,k) = A083323(n).

Sum_{k=0..n} 2^(n-k)*T(n,k) = A122983(n).

G.f.: (1 - 2*x*y - x^2 + x^2*y^2 + x^2*y)/(1 - 3*x*y - x^2 + 3*x^2*y^2 + x^3*y - x^3*y^3). - Philippe Deléham, Nov 09 2013

T(n,k) = 3*T(n-1,k-1) + T(n-2,k) - 3*T(n-2,k-2) - T(n-3,k-1) + T(n-3,k-3), T(0,0) = T(1,1) = T(2,1) = T(2,2) = 1, T(1,0) = T(2,0) = 0, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Nov 09 2013

Example

Triangle begins:
  1;
  0, 1;
  0, 1, 1;
  0, 0, 3,  1;
  0, 1, 0,  6,  1;
  0, 0, 5,  0, 10,   1;
  0, 1, 0, 15,  0,  15,   1;
  0, 0, 7,  0, 35,   0,  21,   1;
  0, 1, 0, 28,  0,  70,   0,  28,  1;
  0, 0, 9,  0, 84,   0, 126,   0, 36,  1;
  0, 1, 0, 45,  0, 210,   0, 210,  0, 45, 1;

Maple

T:= proc(n, k) option remember;
      if k<0 or k>n then 0
    elif k=n then 1
    elif n=2 and k=1 then 1
    elif k=0 then 0
    else 3*T(n-1, k-1) + T(n-2, k) - 3*T(n-2, k-2) - T(n-3, k-1) + T(n-3, k-3)
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..10); # G. C. Greubel, Feb 17 2020

Mathematica

With[{m = 10}, CoefficientList[CoefficientList[Series[(1-2*x*y-x^2+x^2*y^2+
x^2*y)/(1-3*x*y-x^2+3*x^2*y^2+x^3*y-x^3*y^3), {x, 0 , m}, {y, 0, m}], x], y]] // Flatten (* Georg Fischer, Feb 17 2020 *)

Prog

(PARI) T(n, k) = if(k<0 || k>n, 0, if(k==n, 1, if(n==2 && k==1, 1, if(k==0, 0, 3*T(n-1, k-1) + T(n-2, k) - 3*T(n-2, k-2) - T(n-3, k-1) + T(n-3, k-3) )))); \\ G. C. Greubel, Feb 17 2020
(Sage)
@CachedFunction
def T(n, k):
    if (k<0 or k>n): return 0
    elif (k==n): return 1
    elif (n=2 and k=1): then 1
    elif (k=0): then 0
    else: return 3*T(n-1, k-1) + T(n-2, k) - 3*T(n-2, k-2) - T(n-3, k-1) + T(n-3, k-3)
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Feb 17 2020

Crossrefs

Cf. A000004, A000012, A000217, A000332, A000579, A000581, A007318.

Sequence in context: A152892 A193002 A366725 * A242887 A307753 A181116

Adjacent sequences: A122957 A122958 A122959 * A122961 A122962 A122963

Keyword

nonn,tabl

Author

Philippe Deléham, Oct 26 2006

Extensions

a(12) corrected by Georg Fischer, Feb 17 2020

Status

approved