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

0,2

Comments

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

Links

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

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.

Example

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;

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 *)

Prog

(PARI) T(n, k) = if(n<k, 0, if(k<0, 0, if((n==0)&&(k==0), 1, 2*T(n-1, k)+2*T(n-1, k-1)+T(n-2, k)-T(n-2, k-2))));
matrix(10, 10, n, k, T(n-1, k-1)) \\ to see the triangle \\ Michel 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

Crossrefs

Cf. A000027, A000129, A006645, A007290.

Sequence in context: A193906 A224791 A210637 * A202396 A210804 A087910

Adjacent sequences: A201969 A201970 A201971 * A201973 A201974 A201975

Keyword

nonn,tabl

Author

Philippe Deléham, Dec 07 2011

Extensions

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

Status

approved