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 A261780 Number A(n,k) of compositions 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, 4, 0, 1, 4, 15, 24, 8, 0, 1, 5, 26, 73, 82, 16, 0, 1, 6, 40, 164, 354, 280, 32, 0, 1, 7, 57, 310, 1031, 1716, 956, 64, 0, 1, 8, 77, 524, 2395, 6480, 8318, 3264, 128, 0, 1, 9, 100, 819, 4803, 18501, 40728, 40320, 11144, 256, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS Also the number of k-compositions of n: matrices with k rows of nonnegative integers with positive column sums and total element sum n. A(2,2) = 7: (matrices and corresponding marked compositions are given) [1 1] [0 0] [1 0] [0 1] [1] [2] [0] [0 0] [1 1] [0 1] [1 0] [1] [0] [2] 1a1a, 1b1b, 1a1b, 1b1a, 2ab, 2aa, 2bb. LINKS Alois P. Heinz, Antidiagonals n = 0..140, flattened E. Grazzini, E. Munarini, M. Poneti, S. Rinaldi, m-compositions and m-partitions: exhaustive generation and Gray code, Pure Math. Appl. 17 (2006), 111-121. G. Louchard, Matrix Compositions: a Probabilistic analysis, Proc. GASCOM'08, Pure Mathematics and Applications, 19, 2-3, 127-146, 2008. E. Munarini, M. Poneti, S. Rinaldi, Matrix compositions, Journal of Integer Sequences, Vol. 12 (2009), Article 09.4.8 FORMULA G.f. of column k: (1-x)^k/(2*(1-x)^k-1). A(n,k) = Sum_{i=0..k} C(k,i) * A261781(n,k-i). A(n,k) = Sum_{j>=0} (1/2)^(j+1) * binomial(n-1+k*j,n). - Seiichi Manyama, Aug 06 2024 EXAMPLE A(3,2) = 24: 3aaa, 3aab, 3abb, 3bbb, 2aa1a, 2aa1b, 2ab1a, 2ab1b, 2bb1a, 2bb1b, 1a2aa, 1a2ab, 1a2bb, 1b2aa, 1b2ab, 1b2bb, 1a1a1a, 1a1a1b, 1a1b1a, 1a1b1b, 1b1a1a, 1b1a1b, 1b1b1a, 1b1b1b. Square array A(n,k) begins: 1, 1, 1, 1, 1, 1, 1, ... 0, 1, 2, 3, 4, 5, 6, ... 0, 2, 7, 15, 26, 40, 57, ... 0, 4, 24, 73, 164, 310, 524, ... 0, 8, 82, 354, 1031, 2395, 4803, ... 0, 16, 280, 1716, 6480, 18501, 44022, ... 0, 32, 956, 8318, 40728, 142920, 403495, ... MAPLE A:= proc(n, k) option remember; `if`(n=0, 1, add(A(n-j, k)*binomial(j+k-1, k-1), j=1..n)) end: seq(seq(A(n, d-n), n=0..d), d=0..12); MATHEMATICA a[n_, k_] := SeriesCoefficient[(1-x)^k/(2*(1-x)^k-1), {x, 0, n}]; Table[ a[n-k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Feb 07 2017 *) CROSSREFS Columns k=0-10 give: A000007, A011782, A003480, A145839, A145840, A145841, A161434, A261799, A261800, A261801, A261802. Rows n=0-2 give: A000012, A001477, A005449. Main diagonal gives A261783. Cf. A261718 (same for partitions), A261781. Sequence in context: A362125 A261718 A144074 * A124540 A124550 A306024 Adjacent sequences: A261777 A261778 A261779 * A261781 A261782 A261783 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Aug 31 2015 STATUS approved

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Last modified September 15 03:00 EDT 2024. Contains 375931 sequences. (Running on oeis4.)