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A077227 Triangle of compositions of n into exactly k parts each no more than k. 3
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,5
LINKS
FORMULA
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
EXAMPLE
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).
PROG
(PARI) T(n, k)=polcoeff(((1-x^k)/(1-x +x*O(x^n)))^k, n-k)
for(n=1, 12, for(k=1, n, print1(T(n, k), ", ")); print()) \\ Paul D. Hanna, Jan 25 2013
CROSSREFS
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: A303810 A052249 A030528 * A089263 A348951 A369815
KEYWORD
nonn,tabl
AUTHOR
Henry Bottomley, Oct 29 2002
STATUS
approved

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)