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Search: id:A078121
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| A078121 |
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Infinite lower triangular matrix, M, that satisfies [M^2](i,j) = M(i+1,j+1) for all i,j>=0 where [M^n](i,j) denotes the element at row i, column j, of the n-th power of matrix M, with M(0,k)=1 and M(k,k)=1 for all k>=0. |
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24
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| 1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 10, 16, 8, 1, 1, 36, 84, 64, 16, 1, 1, 202, 656, 680, 256, 32, 1, 1, 1828, 8148, 10816, 5456, 1024, 64, 1, 1, 27338, 167568, 274856, 174336, 43680, 4096, 128, 1, 1, 692004, 5866452, 11622976, 8909648, 2794496, 349504
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OFFSET
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0,5
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COMMENT
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M also satisfies: [M^(2k)](i,j) = [M^k](i+1,j+1) for all i,j,k>=0; thus [M^(2^n)](i,j) = M(i+n,j+n) for all n>=0.
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FORMULA
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M(1, j)=A002577(j) (partitions of 2^j into powers of 2), M(j+1, j)=2^j, M(j+2, j)=4^j, M(j+3, j)=A016131(j).
M(n, k) = the coefficient of x^(2^n - 2^(n-k)) in the power series expansion of 1/Product_{j=0..n-k}(1-x^(2^j)) whenever 0<=k<n for all n>0 (conjecture).
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EXAMPLE
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The square of the matrix is the same matrix excluding the first row and column:
[1,_0,_0,0,0]^2=[_1,_0,_0,_0,0]
[1,_1,_0,0,0]___[_2,_1,_0,_0,0]
[1,_2,_1,0,0]___[_4,_4,_1,_0,0]
[1,_4,_4,1,0]___[10,16,_8,_1,0]
[1,10,16,8,1]___[36,84,64,16,1]
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CROSSREFS
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Cf. A078122, A002577, A016131.
Sequence in context: A034372 A155971 A154218 this_sequence A119732 A123521 A123246
Adjacent sequences: A078118 A078119 A078120 this_sequence A078122 A078123 A078124
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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), Nov 18 2002
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