A117334 Triangle, read by rows, where column 0 is [1,-1,-2,-3,...,-n,...] and column k+1 is generated by the binomial transform of column k preceded by a zero (column k includes the k zeros above the main diagonal).

1, -1, 1, -2, 1, 1, -3, -2, 4, 1, -4, -13, 8, 8, 1, -5, -44, -3, 38, 13, 1, -6, -123, -117, 125, 101, 19, 1, -7, -314, -718, 205, 594, 213, 26, 1, -8, -761, -3314, -954, 2787, 1822, 393, 34, 1, -9, -1784, -13481, -12644, 9717, 12987, 4507, 663, 43, 1, -10, -4087, -51055, -90625, 12247, 79419, 43282, 9727, 1048

Offset

0,4

Comments

Row sums are all zeros after row 0.

Links

Table of n, a(n) for n=0..63.

Formula

T(n,k) = Sum_{i=k..n} C(n,i)*T(i-1,k-1) for k>0, with T(0,0)=1 and T(n,0)=-n for n>0.

Example

To generate, start with [1,-1,-2,-3,-4,...] in column 0.
Then column 1 = BINOMIAL[0, 1,-1,-2,-3,-4,-5,...]
= [0,1,1,-2,-13,-44,-123,-314,-761,...];
column 2 = BINOMIAL[0, 0,1,1,-2,-13,-44,-123,-314,...]
= [0,0,1,4,8,-3,-117,-718,-3314,-13481,...];
column 3 = BINOMIAL[0, 0,0,1,4,8,-3,-117,-718,...]
= [0,0,0,1,8,38,125,205,-954,-12644,-90625,...]; etc.
Triangle begins:
1;
-1,1;
-2,1,1;
-3,-2,4,1;
-4,-13,8,8,1;
-5,-44,-3,38,13,1;
-6,-123,-117,125,101,19,1;
-7,-314,-718,205,594,213,26,1;
-8,-761,-3314,-954,2787,1822,393,34,1;
-9,-1784,-13481,-12644,9717,12987,4507,663,43,1;
-10,-4087,-51055,-90625,12247,79419,43282,9727,1048,53,1; ...

Prog

(PARI) T(n, k)=if(n<k || k<0, 0, if(n==k, 1, if(k==0, -n, sum(i=k, n, binomial(n, i)*T(i-1, k-1)))))

Crossrefs

Cf. A117335 (matrix inverse), A117336, A117337, A117338.

Sequence in context: A240853 A363094 A319849 * A241472 A071872 A141038

Adjacent sequences: A117331 A117332 A117333 * A117335 A117336 A117337

Keyword

sign,tabl

Author

Paul D. Hanna, Mar 08 2006

Status

approved