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

0,3

Comments

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

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

Links

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

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).

Example

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;

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A309604 A244009 A321482 * A011415 A216473 A378390

Adjacent sequences: A182039 A182040 A182041 * A182043 A182044 A182045

Keyword

easy,nonn,tabl

Author

Philippe Deléham, Apr 07 2012

Extensions

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

Status

approved