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Search: id:A098447
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| A098447 |
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Triangle T, read by rows, such that diagonal n equals column 0 of T^(n+1), the (n+1)-th matrix power of T. |
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+0
2
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| 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
(list; table; graph; listen)
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OFFSET
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0,5
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COMMENT
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Row sums form A098448.
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FORMULA
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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.
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EXAMPLE
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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.
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PROGRAM
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(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)); ))}
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CROSSREFS
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Cf. A098448, A098446, A091351.
Adjacent sequences: A098444 A098445 A098446 this_sequence A098448 A098449 A098450
Sequence in context: A138028 A009999 A098446 this_sequence A122175 A073165 A137153
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KEYWORD
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nonn,tabl
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Sep 07 2004
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