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A320264 Number T(n,k) of proper multisets of nonempty words with a total of n letters over k-ary alphabet such that all k letters occur at least once in the multiset; triangle T(n,k), n>=2, 1<=k<=n-1, read by rows. 4
1, 1, 2, 3, 11, 9, 4, 38, 84, 52, 7, 125, 523, 766, 365, 10, 364, 2676, 7096, 7775, 3006, 16, 1041, 12435, 52955, 100455, 87261, 28357, 22, 2838, 54034, 348696, 1020805, 1497038, 1074766, 301064, 32, 7645, 225417, 2120284, 8995801, 19823964, 23605043, 14423564, 3549177 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

2,3

LINKS

Alois P. Heinz, Rows n = 2..150

FORMULA

T(n,k) = A257740(n,k) - A319501(n,k).

EXAMPLE

T(2,1) = 1: {a,a}.

T(3,2) = 2: {a,a,b}, {a,b,b}.

T(4,3) = 9: {a,a,b,c}, {a,a,bc}, {a,a,cb}, {b,b,a,c}, {b,b,ac}, {b,b,ca}, {c,c,a,b}, {c,c,ab}, {c,c,ba}.

Triangle T(n,k) begins:

  .

  .   .

  .   1,    .

  .   1,    2,     .

  .   3,   11,     9,      .

  .   4,   38,    84,     52,       .

  .   7,  125,   523,    766,     365,       .

  .  10,  364,  2676,   7096,    7775,    3006,       .

  .  16, 1041, 12435,  52955,  100455,   87261,   28357,      .

  .  22, 2838, 54034, 348696, 1020805, 1497038, 1074766, 301064,   .

MAPLE

h:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,

      add(h(n-i*j, i-1, k)*binomial(k^i, j), j=0..n/i)))

    end:

g:= proc(n, k) option remember; `if`(n=0, 1, add(add(

      d*k^d, d=numtheory[divisors](j))*g(n-j, k), j=1..n)/n)

    end:

T:= (n, k)-> add((-1)^i*(g(n, k-i)-h(n$2, k-i))*binomial(k, i), i=0..k):

seq(seq(T(n, k), k=1..n-1), n=2..12);

CROSSREFS

Column k=1 gives A047967.

Row sums give A320265.

T(n+1,n) gives A006152.

Cf. A257740, A319501.

Sequence in context: A086146 A083758 A127494 * A265563 A265547 A130803

Adjacent sequences:  A320261 A320262 A320263 * A320265 A320266 A320267

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Oct 08 2018

STATUS

approved

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Last modified October 15 04:33 EDT 2019. Contains 328026 sequences. (Running on oeis4.)