A100936 Symmetric square array, read by antidiagonals, where the inverse binomial transform of row n equals: [C(n,0)*1, C(n,1)*2,..., C(n,k)*A051163(k), ..., C(n,n)*A051163(n)] and where A051162 equals the antidiagonal sums.

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 14, 7, 1, 1, 9, 28, 28, 9, 1, 1, 11, 47, 76, 47, 11, 1, 1, 13, 71, 163, 163, 71, 13, 1, 1, 15, 100, 301, 435, 301, 100, 15, 1, 1, 17, 134, 502, 971, 971, 502, 134, 17, 1, 1, 19, 173, 778, 1909, 2577, 1909, 778, 173, 19, 1, 1, 21, 217, 1141

Offset

0,5

Comments

Antidiagonal sums form A051163. Main diagonal is A100937. Different from A086620.

Links

Table of n, a(n) for n=0..69.

Formula

T(n, k) = Sum_{j=0..n} C(k, j)*C(n, j)*A051162(j), with T(0, 0) = 1 and where Sum_{i=0..n} T(n-i, i) = A051162(n).

Example

Rows begin:
[1,1,1,1,1,1,1,1,1,...],
[1,3,5,7,9,11,13,15,17,...],
[1,5,14,28,47,71,100,134,...],
[1,7,28,76,163,301,502,778,...],
[1,9,47,163,435,971,1909,3417,...],
[1,11,71,301,971,2577,5917,12167,...],
[1,13,100,502,1909,5917,15678,36744,...],
[1,15,134,778,3417,12167,36744,97272,...],...
Antidiagonal sums form A051163: [1,2,5,12,30,76,194,496,1269,3250,8337,...].
The inverse binomial transform of the rows form the respective rows of the triangle B:
[1*1],
[1*1,1*2],
[1*1,2*2,1*5],
[1*1,3*2,3*5,1*12],
[1*1,4*2,6*5,4*12,1*30],...
where B(n,k) = binomial(n,k)*A051163(k).

Prog

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

Crossrefs

Cf. A051163, A100937.

Sequence in context: A144461 A106597 A108359 * A086620 A338934 A228356

Adjacent sequences: A100933 A100934 A100935 * A100937 A100938 A100939

Keyword

nonn,tabl

Author

Paul D. Hanna, Nov 23 2004

Status

approved