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Search: id:A059481
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| A059481 |
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Triangle read by rows: T(n,k) = number of ways to distribute k identical objects in n distinct containers; containers may be left empty. |
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+0
6
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| 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, 462, 1, 7, 28, 84, 210, 462, 924, 1716, 1, 8, 36, 120, 330, 792, 1716, 3432, 6435, 1, 9, 45, 165, 495, 1287, 3003, 6435, 12870, 24310, 1, 10, 55, 220, 715, 2002, 5005, 11440
(list; table; graph; listen)
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OFFSET
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1,5
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COMMENT
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Coefficients of Faber polynomials for function x^2/(x-1). - Michael Somos, Sep 09 2003
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REFERENCES
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R. Grimaldi, Discrete and Combinatorial Mathematics, Addison-Wesley, 4:th edition, chapter 1.4.
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FORMULA
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T(n, k)=binomial(n+k-1, k), but triangle includes only n>=k>=0.
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EXAMPLE
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1; 1,1; 1,2,3; 1,3,6,10; 1,4,10,20,35; ...
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.
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).
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MAPLE
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for n from 0 to 20 do for m from 0 to n do printf(`%d, `, binomial(n+m-1, m)) od:od:
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PROGRAM
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(PARI) T(n, k)=if(k<0|k>n, 0, binomial(n+k-1, k))
(PARI) T(n, k)=if(n<0, 0, polcoeff(Pol(((1/(x-x^2)+x*O(x^n))^n+O(x))*x^n), k))
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CROSSREFS
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Take Pascal's triangle A007318, delete entries to the right of a vertical line just right of center, then scan the diagonals.
For a signed version of this triangle see A027555.
Row sums give A000984.
Sequence in context: A111808 A081422 A027555 this_sequence A113592 A136555 A063967
Adjacent sequences: A059478 A059479 A059480 this_sequence A059482 A059483 A059484
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KEYWORD
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easy,nice,nonn,tabl
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AUTHOR
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Fabian Rothelius (fabian.rothelius(AT)telia.com), Feb 04 2001
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EXTENSIONS
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More terms from James A. Sellers (sellersj(AT)math.psu.edu), Feb 07 2001
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