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A077227
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Triangle of compositions of n into exactly k parts each no more than k.
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3
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1, 0, 1, 0, 2, 1, 0, 1, 3, 1, 0, 0, 6, 4, 1, 0, 0, 7, 10, 5, 1, 0, 0, 6, 20, 15, 6, 1, 0, 0, 3, 31, 35, 21, 7, 1, 0, 0, 1, 40, 70, 56, 28, 8, 1, 0, 0, 0, 44, 121, 126, 84, 36, 9, 1, 0, 0, 0, 40, 185, 252, 210, 120, 45, 10, 1, 0, 0, 0, 31, 255, 456, 462, 330, 165, 55, 11, 1, 0, 0, 0, 20
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OFFSET
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1,5
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LINKS
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Table of n, a(n) for n=1..82.
Index entries for sequences related to compositions
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FORMULA
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T(n, k) = A077228(n, k) - A077228(n-1, k).
If n>=k^2, T(n, k) = 0. If k<=n<2k, T(n, k) = C(n-1, k-1).
G.f. of column k is: x^k*(1-x^k)^k/(1-x)^k for k>=1. - Paul D. Hanna, Jan 25 2013
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EXAMPLE
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T(6,3)=7 since 6 can be written as 1+2+3, 1+3+2, 2+1+3, 2+2+2, 2+3+1, 3+1+2, or 3+2+1.
Triangle begins:
1;
0, 1;
0, 2, 1;
0, 1, 3, 1;
0, 0, 6, 4, 1;
0, 0, 7, 10, 5, 1;
0, 0, 6, 20, 15, 6, 1;
0, 0, 3, 31, 35, 21, 7, 1;
0, 0, 1, 40, 70, 56, 28, 8, 1;
0, 0, 0, 44, 121, 126, 84, 36, 9, 1;
0, 0, 0, 40, 185, 252, 210, 120, 45, 10, 1; ...
where column sums are k^k (A000312).
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PROG
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(PARI) {T(n, k)=polcoeff(((1-x^k)/(1-x +x*O(x^n)))^k, n-k)} \\ Paul D. Hanna, Jan 25 2013
for(n=1, 12, for(k=1, n, print1(T(n, k), ", ")); print(""))
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CROSSREFS
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Column sums are A000312. Row sums are A077229. Central diagonal is A000984 offset. Right hand side is right hand side of A007318. Cf. A077228.
Sequence in context: A131185 A052249 A030528 * A089263 A156135 A047265
Adjacent sequences: A077224 A077225 A077226 * A077228 A077229 A077230
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KEYWORD
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nonn,tabl
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AUTHOR
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Henry Bottomley, Oct 29 2002
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STATUS
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approved
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