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Search: id:A126198
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| A126198 |
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Triangle read by rows: T(n,k) (1<=k<=n) = number of compositions of n into parts of size <= k. |
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+0
5
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| 1, 1, 2, 1, 3, 4, 1, 5, 7, 8, 1, 8, 13, 15, 16, 1, 13, 24, 29, 31, 32, 1, 21, 44, 56, 61, 63, 64, 1, 34, 81, 108, 120, 125, 127, 128, 1, 55, 149, 208, 236, 248, 253, 255, 256, 1, 89, 274, 401, 464, 492, 504, 509, 511, 512, 1, 144, 504, 773, 912, 976, 1004, 1016, 1021, 1023, 1024
(list; table; graph; listen)
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OFFSET
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1,3
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REFERENCES
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J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 154-155.
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FORMULA
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G.f. for column k: (x-x^(k+1))/(1-2*x+x^(k+1)). [Riordan]
T(n,3)=A008937(n)-A008937(n-3) for n>=3. T(n,4)=A107066(n-1)-A107066(n-5) for n>=5. T(n,5)=A001949(n+4)-A001949(n-1) for n>=5. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 09 2007
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EXAMPLE
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Triangle begins:
1
1 2
1 3 4
1 5 7 8
1 8 13 15 16
1 13 24 29 31 32
1 21 44 56 61 63 64
Could also be extended to a square array:
1 1 1 1 1 1 1 1 ...
1 2 2 2 2 2 2 2 ...
1 3 4 4 4 4 4 4 ...
1 5 7 8 8 8 8 8 ...
1 8 13 15 16 16 16 ...
1 13 24 29 31 32 32 ...
1 21 44 56 61 63 64 ...
which when read by antidiagonals (downwards) gives A048887.
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MAPLE
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A126198 := proc(n, k) coeftayl( x*(1-x^k)/(1-2*x+x^(k+1)), x=0, n); end: for n from 1 to 11 do for k from 1 to n do printf("%d, ", A126198(n, k)); od; od; - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 09 2007
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CROSSREFS
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Rows are partial sums of rows of A048004. Cf. A048887, A092921 for other versions.
2nd column = Fibonacci numbers, next two columns are A000073, A000078; last three diagonals are 2^n, 2^n-1, 2^n-3.
Cf. A082267.
Sequence in context: A063804 A078753 A119443 this_sequence A055888 A094442 A060642
Adjacent sequences: A126195 A126196 A126197 this_sequence A126199 A126200 A126201
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KEYWORD
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nonn,tabl,nice
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AUTHOR
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njas, Mar 09 2007
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EXTENSIONS
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More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 09 2007
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