A261737 -id:A261737 - cp's OEIS frontend

A261718 Number A(n,k) of partitions of n where each part i is marked with a word of length i over a k-ary alphabet whose letters appear in alphabetical order; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 7, 3, 0, 1, 4, 15, 18, 5, 0, 1, 5, 26, 55, 50, 7, 0, 1, 6, 40, 124, 216, 118, 11, 0, 1, 7, 57, 235, 631, 729, 301, 15, 0, 1, 8, 77, 398, 1470, 2780, 2621, 684, 22, 0, 1, 9, 100, 623, 2955, 8001, 12954, 8535, 1621, 30, 0

Offset: 0

Alois P. Heinz, Aug 29 2015

			A(3,2) = 18: 3aaa, 3aab, 3abb, 3bbb, 2aa1a, 2aa1b, 2ab1a, 2ab1b, 2bb1a, 2bb1b, 1a1a1a, 1a1a1b, 1a1b1a, 1a1b1b, 1b1a1a, 1b1a1b, 1b1b1a, 1b1b1b.
Square array A(n,k) begins:
  1,  1,   1,    1,     1,      1,      1,       1, ...
  0,  1,   2,    3,     4,      5,      6,       7, ...
  0,  2,   7,   15,    26,     40,     57,      77, ...
  0,  3,  18,   55,   124,    235,    398,     623, ...
  0,  5,  50,  216,   631,   1470,   2955,    5355, ...
  0,  7, 118,  729,  2780,   8001,  19158,   40299, ...
  0, 11, 301, 2621, 12954,  45865, 130453,  317905, ...
  0, 15, 684, 8535, 55196, 241870, 820554, 2323483, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-10 give: A000007, A000041, A074141, A261737, A261738, A261739, A261740, A261741, A261742, A261743, A261744.
Rows n=0-2 give: A000012, A001477, A005449.
Main diagonal gives A209668.
Cf. A144064, A261719, A261780.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k)+`if`(i>n, 0, b(n-i, i, k)*binomial(i+k-1, k-1))))
    end:
A:= (n, k)-> b(n, n, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + If[i > n, 0, b[n - i, i, k]*Binomial[i + k - 1, k - 1]]]]; A[n_, k_] := b[n, n, k]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Feb 22 2016, after Alois P. Heinz *)

A(n,k) = Sum_{i=0..k} C(k,i) * A261719(n,k-i).

A293367 Number of partitions of n where each part i is marked with a word of length i over a ternary alphabet whose letters appear in alphabetical order and all three letters occur at least once in the partition.

Original entry on oeis.org

10, 81, 396, 1751, 6528, 23892, 80979, 272085, 876342, 2821217, 8840964, 27713589, 85532512, 263935014, 806417553, 2464692788, 7483544643, 22727335830, 68734242687, 207887123472, 627024671262, 1891376241178, 5694616254570, 17146333061406, 51564199968339

Offset: 3

Alois P. Heinz, Oct 07 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..1000

Column k=3 of A261719.

Cf. A261737.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k)+`if`(i>n, 0, b(n-i, i, k)*binomial(i+k-1, k-1))))
    end:
a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k))(3):
seq(a(n), n=3..30);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + If[i > n, 0, b[n - i, i, k] Binomial[i + k - 1, k - 1]]]];
a[n_] := With[{k = 3}, Sum[b[n, n, k - i] (-1)^i Binomial[k, i], {i, 0, k}]];
a /@ Range[3, 30] (* Jean-François Alcover, Dec 08 2020, after Alois P. Heinz *)

a(n) ~ c * 3^n, where c = 6.846206073498521357898163368676070142316815386135993166380819930419737... - Vaclav Kotesovec, Oct 11 2017

cp's OEIS Frontend

A261718 Number A(n,k) of partitions of n where each part i is marked with a word of length i over a k-ary alphabet whose letters appear in alphabetical order; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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