A098447 Triangle T, read by rows, such that diagonal n equals column 0 of T^(n+1), the (n+1)-th matrix power of T.

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 9, 9, 1, 1, 5, 16, 32, 24, 1, 1, 6, 25, 78, 150, 79, 1, 1, 7, 36, 155, 532, 1018, 340, 1, 1, 8, 49, 271, 1395, 5802, 10996, 2090, 1, 1, 9, 64, 434, 3036, 21343, 116658, 212434, 20613, 1, 1, 10, 81, 652, 5824, 60209, 661325, 5072504

Offset

0,5

Comments

Row sums form A098448.

Links

Table of n, a(n) for n=0..62.

Formula

T(n, k) = Sum_{i=0..k} T(k, i)*T(n-k+i-1, i), for 0<k<n, else T(0, n)=T(n, n)=1.

Example

T(7,3) = T(3,0)*T(3,0) + T(3,1)*T(4,1) + T(3,2)*T(5,2) + T(3,3)*T(6,3)
= 1*1 + 3*4 + 4*16 + 1*78 = 155.
Rows of T begin:
[1],
[1,1],
[1,2,1],
[1,3,4,1],
[1,4,9,9,1],
[1,5,16,32,24,1],
[1,6,25,78,150,79,1],
[1,7,36,155,532,1018,340,1],
[1,8,49,271,1395,5802,10996,2090,1],
[1,9,64,434,3036,21343,116658,212434,20613,1],...
Matrix square T^2 begins:
[1],
[2,1],
[4,4,1],
[9,14,8,1],
[24,53,54,18,1],
[79,234,376,280,48,1],
[340,1291,2976,4034,2196,158,1],...
where column 0 is {1,2,4,9,24,79,340,...} and forms diagonal 1 of T.
Matrix cube T^3 begins:
[1],
[3,1],
[9,6,1],
[32,33,12,1],
[150,219,135,27,1],
[1018,2023,1944,744,72,1],...
where column 0 is {1,3,9,32,150,1018,...} and forms diagonal 2 of T.

Prog

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

Crossrefs

Cf. A098448, A098446, A091351.

Sequence in context: A322268 A144823 A098446 * A202784 A175105 A162717

Adjacent sequences: A098444 A098445 A098446 * A098448 A098449 A098450

Keyword

nonn,tabl

Author

Paul D. Hanna, Sep 07 2004

Status

approved