A220399 A convolution triangle of numbers obtained from A057682.

1, 0, 1, 0, 2, 1, 0, 3, 4, 1, 0, 3, 10, 6, 1, 0, 0, 18, 21, 8, 1, 0, -9, 21, 53, 36, 10, 1, 0, -27, 0, 99, 116, 55, 12, 1, 0, -54, -81, 117, 286, 215, 78, 14, 1, 0, -81, -270, -27, 528, 650, 358, 105, 16, 1

Offset

0,5

Comments

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

Links

Table of n, a(n) for n=0..54.

Formula

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

G.f for k-th column: ((x-x^2)/(1-3*x+3*x^2))^k.

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

Sum_{k, 0<=k<=n, n>0} T(n,k) = A001792(n-1).

T(n+1,n) = 2*n = A005843(n).

T(n+2,n) = A014105(n).

T(n,1) = A057682(n).

Example

Triangle begins :
1
0, 1
0, 2, 1
0, 3, 4, 1
0, 3, 10, 6, 1
0, 0, 18, 21, 8, 1
0, -9, 21, 53, 36, 10, 1
0, -27, 0, 99, 116, 55, 12, 1

Crossrefs

Cf. A030523, A057682, A104597

Sequence in context: A337804 A357368 A193401 * A268830 A095884 A342240

Adjacent sequences: A220396 A220397 A220398 * A220400 A220401 A220402

Keyword

sign,tabl

Author

Philippe Deléham, Feb 19 2013

Status

approved