A096589 Symmetric square array T(n,k)=T(k,n), read by antidiagonals, such that T(n,k) equals the dot product of the k-th antidiagonal with the initial terms of the (n-k)-th row when n>=k, with T(n,0)=1.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 4, 4, 1, 1, 5, 8, 8, 5, 1, 1, 6, 11, 8, 11, 6, 1, 1, 7, 17, 20, 20, 17, 7, 1, 1, 8, 22, 30, 14, 30, 22, 8, 1, 1, 9, 30, 45, 42, 42, 45, 30, 9, 1, 1, 10, 37, 69, 72, 28, 72, 69, 37, 10, 1, 1, 11, 47, 100, 101, 98, 98, 101, 100, 47, 11, 1, 1, 12, 56, 133, 159

Offset

0,5

Comments

Main diagonal equals the antidiagonal sums (A096590).

Links

Table of n, a(n) for n=0..82.

Formula

T(n, k) = Sum_{j=0, k} T(k-j, j)*T(n-k, j) when n>=k, else T(n, k)=T(k, n).

Example

T(5,2) = 17 = 1*1+2*4+1*8 = T(2,0)*T(3,0) + T(1,1)*T(3,1) +
T(0,2)*T(3,2).
T(7,3) = 69 = 1*1+3*5+3*11+1*20 = T(3,0)*T(4,0) + T(2,1)*T(4,1) +
T(1,2)*T(4,2) + T(0,3)*T(4,3).
Rows begin:
[1,1,1,1,1,1,1,1,1,1,...],
[1,2,3,4,5,6,7,8,9,10,...],
[1,3,4,8,11,17,22,30,37,47,...],
[1,4,8,8,20,30,45,69,100,133,...],
[1,5,11,20,14,42,72,101,159,255,...],
[1,6,17,30,42,28,98,184,279,386,...],
[1,7,22,45,72,98,44,176,372,622,...],
[1,8,30,69,101,184,176,90,405,943,...],
[1,9,37,100,159,279,372,405,136,680,...],
[1,10,47,133,255,386,622,943,680,254,...],...

Prog

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

Crossrefs

Cf. A096590.

Sequence in context: A318274 A329330 A049695 * A176427 A324592 A099573

Adjacent sequences: A096586 A096587 A096588 * A096590 A096591 A096592

Keyword

nonn,tabl

Author

Paul D. Hanna, Jun 28 2004

Status

approved