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
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
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(n
-
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
Comments