A123585 Triangle T(n,k), 0<=k<=n, given by [1, -1, 1, 0, 0, 0, 0, 0, ...] DELTA [1, 1, -1, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

1, 1, 1, 0, 2, 2, -1, 1, 5, 3, -1, -2, 4, 10, 5, 0, -4, -4, 12, 20, 8, 1, -2, -13, -4, 31, 38, 13, 1, 3, -11, -33, 3, 73, 71, 21, 0, 6, 6, -42, -74, 34, 162, 130, 34, -1, 3, 24, 0, -130, -146, 128, 344, 235, 55, -1, -4, 21, 72, -50, -352

Offset

0,5

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Formula

Sum_{k,0<=k<=n} T(n,k) = 2^n = A000079(n).

T(n,0) = A010892(n).

T(n,n) = Fibonacci(n+1) = A000045(n+1).

T(n+1,1) = A099254(n).

T(n+1,n) = A001629(n+2).

Sum_{k, 0<=k<=[n/2]} T(n-k,k) = A003269(n).

T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k-2) - T(n-2,k), n>0.

Sum_{k, 0<=k<=n} x^k*T(n,k) = (-1)^n*A003683(n+1), (-1)^n*A006130(n), A000007(n), A010892(n), A000079(n), A030195(n+1) for x=-3, -2, -1, 0, 1, 2 respectively . - Philippe Deléham, Dec 01 2006

T(n+2,n) = A129707(n+1).- Philippe Deléham, Dec 18 2011

G.f.: 1/(1-(1+y)*x+(1-y^2)*x^2). - Philippe Deléham, Dec 18 2011

Example

Triangle begins:
1;
1, 1;
0, 2, 2;
-1, 1, 5, 3;
-1, -2, 4, 10, 5;
0, -4, -4, 12, 20, 8;
1, -2, -13, -4, 31, 38, 13;
1, 3, -11, -33, 3, 73, 71, 21;
0, 6, 6, -42, -74, 34, 162, 130, 34;

Mathematica

CoefficientList[CoefficientList[Series[1/(1 - (1 + y)*x + (1 - y^2)*x^2), {x, 0, 10}, {y, 0, 10}], x], y] // Flatten (* G. C. Greubel, Oct 16 2017 *)

Crossrefs

Cf. A010892, A099254, A000045, A000079 (row sums), A001629, A129707.

Sequence in context: A178655 A337278 A178304 * A145668 A318405 A181645

Adjacent sequences: A123582 A123583 A123584 * A123586 A123587 A123588

Keyword

sign,tabl

Author

Philippe Deléham, Nov 13 2006

Status

approved