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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. 14
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

LINKS

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

FORMULA

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

EXAMPLE

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, ...

MAPLE

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);

MATHEMATICA

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-Fran├žois Alcover, Feb 22 2016, after Alois P. Heinz *)

CROSSREFS

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.

Sequence in context: A294498 A292860 A265609 * A144074 A261780 A124540

Adjacent sequences:  A261715 A261716 A261717 * A261719 A261720 A261721

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Aug 29 2015

STATUS

approved

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Last modified October 23 19:37 EDT 2019. Contains 328373 sequences. (Running on oeis4.)