A182059 Triangle, read by rows, given by (0, 2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

1, 0, 2, 0, 4, 4, 0, 6, 12, 8, 0, 8, 24, 32, 16, 0, 10, 40, 80, 80, 32, 0, 12, 60, 160, 240, 192, 64, 0, 14, 84, 280, 560, 672, 448, 128, 0, 16, 112, 448, 1120, 1792, 1792, 1024, 256, 0, 18, 144, 672, 2016, 4032, 5376, 4608, 2304, 512

Offset: 0

Author

Philippe Deléham, Apr 09 2012

Keywords

• easy
• nonn
• tabl

Comments

Row sums are 3^n - 1 + 0^n.

Triangle of coefficients in expansion of (1+2*x)^n - 1 + 0^n .

Examples

			Triangle begins :
1
0, 2
0, 4, 4
0, 6, 12, 8
0, 8, 24, 32, 16
0, 10, 40, 80, 80, 32
0, 12, 60, 160, 240, 192, 64
0, 14, 84, 280, 560, 672, 448, 128
0, 16, 112, 448, 1120, 1792, 1792, 1024, 256
0, 18, 144, 672, 2016, 4032, 5376, 4608, 2304, 512
0, 20, 180, 960, 3360, 8064, 13440, 15360, 11520, 5120, 1024
		

Crossrefs

Cf. A000079, A007318, A013609, A193789, A193790

Formula

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

T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) - 2*T(n-2,k-1), T(0,0) = 1, T(1,0) = T(2,0) = 0, T(1,1) = 2, T(2,1) = T(2,2) = 4 and T(n,k) = 0 if k<0 or if k>n.

T(n,k) = A206735(n,k)*2^k.

T(n,k) = A013609(n,k) - A073424(n,k) .