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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-François 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: A292860 A265609 A362125 * 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 June 18 20:38 EDT 2024. Contains 373487 sequences. (Running on oeis4.)