A201972 Triangle T(n,k), read by rows, given by (2,1/2,-1/2,0,0,0,0,0,0,0,...) DELTA (2,-1/2,1/2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

1, 2, 2, 5, 8, 3, 12, 28, 20, 4, 29, 88, 94, 40, 5, 70, 262, 372, 244, 70, 6, 169, 752, 1333, 1184, 539, 112, 7, 408, 2104, 4472, 5016, 3144, 1064, 168, 8, 985, 5776, 14316, 19408, 15526, 7344, 1932, 240, 9

Offset: 0

Author

Philippe Deléham, Dec 07 2011

Keywords

• nonn
• tabl

Comments

Diagonal sums: A201967(n), row sums: A000302(n) (powers of 4).

Examples

			Triangle begins:
    1;
    2,   2;
    5,   8,    3;
   12,  28,   20,    4;
   29,  88,   94,   40,   5;
   70, 262,  372,  244,  70,   6;
  169, 752, 1333, 1184, 539, 112, 7;
		

Links

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

Crossrefs

Cf. A000027, A000129, A006645, A007290.

Programs

Maple

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

Mathematica

With[{m = 8}, CoefficientList[CoefficientList[Series[1/(1-2*(y+1)*x+(y+1)*(y-1)*x^2), {x, 0 , m}, {y, 0, m}], x], y]] // Flatten (* Georg Fischer, Feb 17 2020 *)

PARI

T(n,k) = if(nMichel Marcus, Feb 17 2020

Sage

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

Formula

G.f.: 1/(1-2*(y+1)*x+(y+1)*(y-1)*x^2).

Sum_{k=0..n} T(n,k)*x^k = A000007(n), A000129(n+1), A000302(n), A138395(n), A057084(n) for x = -1, 0, 1, 2, 3, respectively.

Sum_{k=0..n} T(n,k)*x^(n-k) = A000027(n), A000302(n), A090018(n), A057091(n) for x = 0, 1, 2, 3, respectively.

T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) - T(n-2,k-2) with T(0,0) = 1, T(n,k) = 0 if k < 0 or if n < k.

Extensions

a(40) corrected by Georg Fischer, Feb 17 2020