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 A243081 Number A(n,k) of compositions of n into parts with multiplicity not larger than k; square array A(n,k), n>=0, k>=0, read by antidiagonals. 21
 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 3, 0, 1, 1, 2, 3, 3, 0, 1, 1, 2, 4, 7, 5, 0, 1, 1, 2, 4, 7, 11, 11, 0, 1, 1, 2, 4, 8, 15, 21, 13, 0, 1, 1, 2, 4, 8, 15, 26, 34, 19, 0, 1, 1, 2, 4, 8, 16, 31, 52, 59, 27, 0, 1, 1, 2, 4, 8, 16, 31, 57, 93, 114, 57, 0, 1, 1, 2, 4, 8, 16, 32, 63, 114, 173, 178, 65, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,13 COMMENTS A(n,k) is the number of compositions of n avoiding the pattern {1}^(k+1). LINKS Alois P. Heinz, Rows n = 0..140, flattened FORMULA A(n,k) = Sum_{i=0..k} A242447(n,i). EXAMPLE Square array A(n,k) begins:   1,  1,  1,  1,   1,   1,   1,   1,   1, ...   0,  1,  1,  1,   1,   1,   1,   1,   1, ...   0,  1,  2,  2,   2,   2,   2,   2,   2, ...   0,  3,  3,  4,   4,   4,   4,   4,   4, ...   0,  3,  7,  7,   8,   8,   8,   8,   8, ...   0,  5, 11, 15,  15,  16,  16,  16,  16, ...   0, 11, 21, 26,  31,  31,  32,  32,  32, ...   0, 13, 34, 52,  57,  63,  63,  64,  64, ...   0, 19, 59, 93, 114, 120, 127, 127, 128, ... MAPLE b:= proc(n, i, p, k) option remember; `if`(n=0, p!, `if`(i<1, 0,       add(b(n-i*j, i-1, p+j, k)/j!, j=0..min(n/i, k))))     end: A:= (n, k)-> `if`(k>=n, `if`(n=0, 1, 2^(n-1)), b(n\$2, 0, k)): seq(seq(A(n, d-n), n=0..d), d=0..14); MATHEMATICA b[n_, i_, p_, k_] := b[n, i, p, k] = If[n == 0, p!, If[i<1, 0, Sum[b[n-i*j, i-1, p+j, k]/j!, {j, 0, Min[n/i, k]}]]]; A[n_, k_] := If[k >= n, If[n == 0, 1, 2^(n-1)], b[n, n, 0, k]]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Feb 02 2015, after Alois P. Heinz *) CROSSREFS Columns k=0-10 give: A000007, A032020, A232432, A232464, A243082, A243083, A243084, A243085, A243086, A243087, A243088. Main diagonal gives A011782. A(2n,n) gives A232605. Sequence in context: A103498 A030386 A096799 * A287847 A336201 A271369 Adjacent sequences:  A243078 A243079 A243080 * A243082 A243083 A243084 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, May 29 2014 STATUS approved

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Last modified August 8 11:31 EDT 2020. Contains 336298 sequences. (Running on oeis4.)