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A309992 Triangle T(n,k) whose n-th row lists in increasing order the multinomial coefficients M(n;lambda), where lambda ranges over all partitions of n into distinct parts; n >= 0, 1 <= k <= A000009(n), read by rows. 5
1, 1, 1, 1, 3, 1, 4, 1, 5, 10, 1, 6, 15, 60, 1, 7, 21, 35, 105, 1, 8, 28, 56, 168, 280, 1, 9, 36, 84, 126, 252, 504, 1260, 1, 10, 45, 120, 210, 360, 840, 1260, 2520, 12600, 1, 11, 55, 165, 330, 462, 495, 1320, 2310, 4620, 6930, 27720 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

First row with repeated terms is row 15, see also A309999: 1365 = M(15;11,4) = M(15;12,2,1) and 30030 = M(15;9,5,1) = M(15;10,3,2).

LINKS

Alois P. Heinz, Rows n = 0..45, flattened

Wikipedia, Multinomial coefficients

Wikipedia, Partition (number theory)

EXAMPLE

For n = 5 there are 3 partitions of 5 into distinct parts: [5], [4,1], [3,2].  So row 5 contains M(5;5) = 1, M(5;4,1) = 5 and M(5;3,2) = 10.

Triangle T(n,k) begins:

  1;

  1;

  1;

  1,  3;

  1,  4;

  1,  5, 10;

  1,  6, 15,  60;

  1,  7, 21,  35, 105;

  1,  8, 28,  56, 168, 280;

  1,  9, 36,  84, 126, 252, 504, 1260;

  1, 10, 45, 120, 210, 360, 840, 1260, 2520, 12600;

  1, 11, 55, 165, 330, 462, 495, 1320, 2310,  4620, 6930, 27720;

  ...

MAPLE

g:= proc(n, i) option remember; `if`(i*(i+1)/2<n, [], `if`(n=0, [1],

     [map(x->binomial(n, i)*x, g(n-i, min(n-i, i-1)))[], g(n, i-1)[]]))

    end:

T:= n-> sort(g(n$2))[]:

seq(T(n), n=0..14);

CROSSREFS

Columns k=1-3 give: A000012, A000027 (for n>=3), A000217(n-1) (for n>=5).

Row sums give A007837.

Rightmost terms of rows give A290517.

Cf. A000009, A036038, A309999, A325901, A325903.

Sequence in context: A187079 A086183 A014462 * A016474 A069264 A064575

Adjacent sequences:  A309989 A309990 A309991 * A309993 A309994 A309995

KEYWORD

nonn,tabf

AUTHOR

Alois P. Heinz, Aug 26 2019

STATUS

approved

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Last modified January 23 01:30 EST 2020. Contains 331166 sequences. (Running on oeis4.)