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A078536
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Infinite lower triangular matrix, M, that satisfies [M^4](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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10
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1, 1, 1, 1, 4, 1, 1, 28, 16, 1, 1, 524, 496, 64, 1, 1, 29804, 41136, 8128, 256, 1, 1, 5423660, 10272816, 2755264, 130816, 1024, 1, 1, 3276048300, 8220685104, 2804672704, 178301696, 2096128, 4096, 1, 1, 6744720496300, 21934062166320
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| M also satisfies: [M^(4k)](i,j) = [M^k](i+1,j+1) for all i,j,k>=0; thus [M^(4^n)](i,j) = M(i+n,j+n) for all n>=0. Conjecture: sum of the n-th row equals the partitions of 4^n into powers of 4.
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FORMULA
| M(n, k) = the coefficient of x^(4^n - 4^(n-k)) in the power series expansion of 1/Product_{j=0..n-k}(1-x^(4^j)) whenever 0<=k<n for all n>0 (conjecture).
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EXAMPLE
| The 4-th power of matrix is the same matrix excluding the first row and column:
[1,__0,__0,_0,0]^4=[____1,____0,___0,__0,0]
[1,__1,__0,_0,0]___[____4,____1,___0,__0,0]
[1,__4,__1,_0,0]___[___28,___16,___1,__0,0]
[1,_28,_16,_1,0]___[__524,__496,__64,__1,0]
[1,524,496,64,1]___[29804,41136,8128,256,1]
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CROSSREFS
| Cf. A078121, A078122, A078537.
Sequence in context: A136449 A140805 A113370 * A173918 A174412 A177939
Adjacent sequences: A078533 A078534 A078535 * A078537 A078538 A078539
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KEYWORD
| nonn,tabl
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Nov 29 2002
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