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

Author

Philippe Deléham, Oct 26 2006

Keywords

• nonn
• tabl

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).

Examples

			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;
		

Links

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

Crossrefs

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

Programs

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 *)

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): return 1
    elif (k==0): return 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)
print([[T(n, k) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Feb 17 2020

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

Extensions

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