A077227 - OEIS

%I #13 Jul 19 2016 10:51:05

%S 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,

%T 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,

%U 185,252,210,120,45,10,1,0,0,0,31,255,456,462,330,165,55,11,1,0,0,0,20

%N Triangle of compositions of n into exactly k parts each no more than k.

%H <a href="/index/Com#comp">Index entries for sequences related to compositions</a>

%F T(n, k) = A077228(n, k) - A077228(n-1, k).

%F If n>=k^2, T(n, k) = 0. If k<=n<2k, T(n, k) = C(n-1, k-1).

%F G.f. of column k is: x^k*(1-x^k)^k/(1-x)^k for k>=1. - _Paul D. Hanna_, Jan 25 2013

%e 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.

%e Triangle begins:

%e 1;

%e 0, 1;

%e 0, 2, 1;

%e 0, 1, 3, 1;

%e 0, 0, 6, 4, 1;

%e 0, 0, 7, 10, 5, 1;

%e 0, 0, 6, 20, 15, 6, 1;

%e 0, 0, 3, 31, 35, 21, 7, 1;

%e 0, 0, 1, 40, 70, 56, 28, 8, 1;

%e 0, 0, 0, 44, 121, 126, 84, 36, 9, 1;

%e 0, 0, 0, 40, 185, 252, 210, 120, 45, 10, 1; ...

%e where column sums are k^k (A000312).

%o (PARI) T(n,k)=polcoeff(((1-x^k)/(1-x +x*O(x^n)))^k,n-k)

%o for(n=1,12,for(k=1,n,print1(T(n,k),", "));print()) \\ _Paul D. Hanna_, Jan 25 2013

%Y Column sums are A000312. Row sums are A077229. Central diagonal is A000984 offset. Right hand side is right hand side of A007318. Cf. A077228.

%K nonn,tabl

%O 1,5

%A _Henry Bottomley_, Oct 29 2002