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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); MATHEMATICA h[n_, i_, k_] := h[n, i, k] = If[n == 0, 1, If[i<1, 0,      Sum[h[n - i*j, i-1, k]*Binomial[k^i, j], {j, 0, n/i}]]]; g[n_, k_] := g[n, k] = If[n == 0, 1, Sum[Sum[      d*k^d, {d, Divisors[j]}]*g[n - j, k], {j, 1, n}]/n]; T[n_, k_] := Sum[(-1)^i*(g[n, k-i]-h[n, n, k-i])*Binomial[k, i], {i, 0, k}]; Table[Table[T[n, k], {k, 1, n - 1}], {n, 2, 12}] // Flatten (* Jean-François Alcover, Feb 13 2021, after Alois P. Heinz *) 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 * A335173 A265563 A265547 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 May 9 10:07 EDT 2021. Contains 343720 sequences. (Running on oeis4.)