Search: id:A059481
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%I A059481
%S A059481 1,1,1,1,2,3,1,3,6,10,1,4,10,20,35,1,5,15,35,70,126,1,6,21,56,126,252,
%T A059481 462,1,7,28,84,210,462,924,1716,1,8,36,120,330,792,1716,3432,6435,1,9,
%U A059481 45,165,495,1287,3003,6435,12870,24310,1,10,55,220,715,2002,5005,11440
%N A059481 Triangle read by rows: T(n,k) = number of ways to distribute k identical 
               objects in n distinct containers; containers may be left empty.
%C A059481 Coefficients of Faber polynomials for function x^2/(x-1). - Michael Somos, 
               Sep 09 2003
%D A059481 R. Grimaldi, Discrete and Combinatorial Mathematics, Addison-Wesley, 
               4:th edition, chapter 1.4.
%F A059481 T(n, k)=binomial(n+k-1, k), but triangle includes only n>=k>=0.
%e A059481 1; 1,1; 1,2,3; 1,3,6,10; 1,4,10,20,35; ...
%e A059481 T(3,3) = 10 because the ways to distribute the 3 objects into the three 
               containers are: (3,0,0) (0,3,0) (0,0,3) (2,1,0) (1,2,0) (2,0,1) (1,
               0,2) (0,1,2) (0,2,1) (1,1,1), for a total of 10 possibilities.
%e A059481 T(3,3)=10 since (x^2/(x-1))^3 = (x+1+1/x+O(1/x^2))^3 = x^3+3x^2+6x+10+O(x).
%p A059481 for n from 0 to 20 do for m from 0 to n do printf(`%d,`,binomial(n+m-1,
               m)) od:od:
%o A059481 (PARI) T(n,k)=if(k<0|k>n,0,binomial(n+k-1,k))
%o A059481 (PARI) T(n,k)=if(n<0,0,polcoeff(Pol(((1/(x-x^2)+x*O(x^n))^n+O(x))*x^n),
               k))
%Y A059481 Take Pascal's triangle A007318, delete entries to the right of a vertical 
               line just right of center, then scan the diagonals.
%Y A059481 For a signed version of this triangle see A027555.
%Y A059481 Row sums give A000984.
%Y A059481 Sequence in context: A111808 A081422 A027555 this_sequence A158498 A113592 
               A136555
%Y A059481 Adjacent sequences: A059478 A059479 A059480 this_sequence A059482 A059483 
               A059484
%K A059481 easy,nice,nonn,tabl
%O A059481 1,5
%A A059481 Fabian Rothelius (fabian.rothelius(AT)telia.com), Feb 04 2001
%E A059481 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Feb 07 2001

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