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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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7
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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; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| Also has an interpretation as number of binary vectors of length n-1 in which the length of the longest run of 1's is <= k=1 (see A048004). - N. J. A. Sloane, Apr 03 2011
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REFERENCES
| J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 154-155.
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LINKS
| Alois P. Heinz, Rows n = 1..141
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FORMULA
| 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, Mar 09 2007
T(n,k) = A181695(n,k) - A181695(n-1,k). [From Max Alekseyev, Nov 18 2010]
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EXAMPLE
| 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
| 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, Mar 09 2007
##
T:= proc(n, k) option remember;
if n=0 or k=1 then 1
else add (T(n-j, k), j=1..min(n, k))
fi
end:
seq (seq (T(n, k), k=1..n), n=1..15); # Alois P. Heinz, Oct 23 2011
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MATHEMATICA
| rows = 11; t[n_, k_] := Sum[ (-1)^i*2^(n-i*(k+1))*Binomial[ n-i*k, i], {i, 0, Floor[n/(k+1)]}] - Sum[ (-1)^i*2^((-i)*(k+1)+n-1)*Binomial[ n-i*k-1, i], {i, 0, Floor[(n-1)/(k+1)]}]; Flatten[ Table[ t[n, k], {n, 1, rows}, {k, 1, n}]](* From Jean-François Alcover, Nov 17 2011, after Max Alekseyev *)
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CROSSREFS
| 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 * A055888 A094442 A060642
Adjacent sequences: A126195 A126196 A126197 * A126199 A126200 A126201
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KEYWORD
| nonn,tabl,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Mar 09 2007
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EXTENSIONS
| More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 09 2007
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