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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 A156135 A047265 Adjacent sequences:  A077224 A077225 A077226 * A077228 A077229 A077230 KEYWORD nonn,tabl AUTHOR Henry Bottomley, Oct 29 2002 STATUS approved

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Last modified October 20 15:08 EDT 2019. Contains 328267 sequences. (Running on oeis4.)