A182042 Triangle T(n,k), read by rows, given by (0, 2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (3, 0, -3/2, 3/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

1, 0, 3, 0, 6, 9, 0, 9, 27, 27, 0, 12, 54, 108, 81, 0, 15, 90, 270, 405, 243, 0, 18, 135, 540, 1215, 1458, 729, 0, 21, 189, 945, 2835, 5103, 5103, 2187, 0, 24, 252, 1512, 5670, 13608, 20412, 17496, 6561, 0, 27, 324, 2268, 10206, 30618, 61236, 78732, 59049, 19683

Offset: 0

Author

Philippe Deléham, Apr 07 2012

Keywords

• easy
• nonn
• tabl

Comments

Row sums are 4^n - 1 + 0^n.

Triangle of coefficients in expansion of (1+3*x)^n - 1 + 0^n.

Examples

			Triangle begins:
  1;
  0,  3;
  0,  6,   9;
  0,  9,  27,  27;
  0, 12,  54, 108,   81;
  0, 15,  90, 270,  405,  243;
  0, 18, 135, 540, 1215, 1458,  729;
  0, 21, 189, 945, 2835, 5103, 5103, 2187;
		

Links

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

Crossrefs

Cf. A000244, A007318, A013610, A193193, A193194.

Programs

Maple

T:= proc(n, k) option remember;
      if k=n then 3^n
    elif k=0 then 0
    else binomial(n,k)*3^k
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..10); # G. C. Greubel, Feb 17 2020

Mathematica

With[{m = 9}, CoefficientList[CoefficientList[Series[(1-2*x+x^2+3*y*x^2)/(1-2*x-3*y*x+x^2+3*y*x^2), {x, 0 , m}, {y, 0, m}], x], y]] // Flatten (* Georg Fischer, Feb 17 2020 *)

PARI

T(n,k) = if (k==0, 1, binomial(n,k)*3^k);
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==n): return 3^n
    elif (k==0): return 0
    else: return binomial(n,k)*3^k
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Feb 17 2020

Formula

T(n,0) = 0^n; T(n,k) = binomial(n,k)*3^k for k > 0.

G.f.: (1-2*x+x^2+3*y*x^2)/(1-2*x-3*y*x+x^2+3*y*x^2).

T(n,k) = 2*T(n-1,k) + 3*T(n-1,k-1) - T(n-2,k) -3*T(n-2,k-1), T(0,0) = 1, T(1,0) = T(2,0) = 0, T(1,1) = 3, T(2,1) = 6, T(2,2) = 9 and T(n,k) = 0 if k < 0 or if k > n.

T(n,k) = A206735(n,k)*3^k.

T(n,k) = A013610(n,k) - A073424(n,k).

Extensions

a(48) corrected by Georg Fischer, Feb 17 2020