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A091351
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Triangle T, read by rows, such that T(n,k) equals the (n-k)-th row sum of T^k, where T^k is the k-th power of T as a lower triangular matrix.
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21
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1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 9, 9, 4, 1, 1, 24, 30, 16, 5, 1, 1, 77, 115, 70, 25, 6, 1, 1, 295, 510, 344, 135, 36, 7, 1, 1, 1329, 2602, 1908, 805, 231, 49, 8, 1, 1, 6934, 15133, 11904, 5325, 1616, 364, 64, 9, 1, 1, 41351, 99367, 83028, 39001, 12381, 2919, 540, 81, 10, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Since T(n,0)=1 for n>=0, then the k-th column of the lower triangular matrix T equals the left-most column of T^(k+1) for k>=0.
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FORMULA
| T(n, k) = sum_{j=0..n-k} T(n-k, j)*T(j+k-1, k-1) for n>=k>0 with T(n, 0)=1 (n>=0).
Equals SHIFT_UP(A104445), or A104445(n+1, k) = T(n, k) for n>=k>=0, where triangular matrix X=A104445 satisfies: SHIFT_LEFT_UP(X) = X^2 - X + I.
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EXAMPLE
| T(7,3) = 344 = 1*1 + 9*3 + 9*9 + 4*30 + 1*115
= T(4,0)*T(2,2) +T(4,1)*T(3,2) +T(4,2)*T(4,2) +T(4,3)*T(5,2) +T(4,4)*T(6,2).
Rows begin:
{1},
{1,1},
{1,2,1},
{1,4,3,1},
{1,9,9,4,1},
{1,24,30,16,5,1},
{1,77,115,70,25,6,1},
{1,295,510,344,135,36,7,1},
{1,1329,2602,1908,805,231,49,8,1},
{1,6934,15133,11904,5325,1616,364,64,9,1},...
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PROG
| (PARI) {T(n, k)=if(k>n|n<0|k<0, 0, if(k==0|k==n, 1, sum(j=0, n-k, T(n-k, j)*T(j+k-1, k-1)); ); )}
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CROSSREFS
| Cf. A091352, A091353, A091354.
Cf. A104445.
Sequence in context: A101494 A125781 A091150 * A058730 A112705 A070895
Adjacent sequences: A091348 A091349 A091350 * A091352 A091353 A091354
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KEYWORD
| nonn,tabl
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Jan 02 2004
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