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 A327622 Number A(n,k) of parts in all k-times partitions of n into distinct parts; square array A(n,k), n>=0, k>=0, read by antidiagonals. 7
 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 3, 1, 0, 1, 1, 5, 3, 1, 0, 1, 1, 7, 8, 5, 1, 0, 1, 1, 9, 16, 15, 8, 1, 0, 1, 1, 11, 27, 35, 28, 10, 1, 0, 1, 1, 13, 41, 69, 73, 49, 13, 1, 0, 1, 1, 15, 58, 121, 160, 170, 86, 18, 1, 0, 1, 1, 17, 78, 195, 311, 460, 357, 156, 25, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,14 COMMENTS Row n is binomial transform of the n-th row of triangle A327632. LINKS Alois P. Heinz, Antidiagonals n = 0..200, flattened Wikipedia, Partition (number theory) FORMULA A(n,k) = Sum_{i=0..k} binomial(k,i) * A327632(n,i). EXAMPLE Square array A(n,k) begins: 0, 0, 0, 0, 0, 0, 0, 0, 0, ... 1, 1, 1, 1, 1, 1, 1, 1, 1, ... 1, 1, 1, 1, 1, 1, 1, 1, 1, ... 1, 3, 5, 7, 9, 11, 13, 15, 17, ... 1, 3, 8, 16, 27, 41, 58, 78, 101, ... 1, 5, 15, 35, 69, 121, 195, 295, 425, ... 1, 8, 28, 73, 160, 311, 553, 918, 1443, ... 1, 10, 49, 170, 460, 1047, 2106, 3865, 6611, ... 1, 13, 86, 357, 1119, 2893, 6507, 13182, 24625, ... MAPLE b:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(k=0, [1, 1], `if`(i*(i+1)/2 (f-> f +[0, f[1]*h[2]/h[1]])(h[1]* b(n-i, min(n-i, i-1), k)))(b(i\$2, k-1))))) end: A:= (n, k)-> b(n\$2, k)[2]: seq(seq(A(n, d-n), n=0..d), d=0..14); MATHEMATICA b[n_, i_, k_] := b[n, i, k] = With[{}, If[n==0, Return@{1, 0}]; If[k == 0, Return@{1, 1}]; If[i(i + 1)/2 < n, Return@{0, 0}]; b[n, i - 1, k] + Function[h, Function[f, f + {0, f[[1]] h[[2]]/h[[1]]}][h[[1]] b[n - i, Min[n - i, i - 1], k]]][b[i, i, k - 1]]]; A[n_, k_] := b[n, n, k][[2]]; Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jun 03 2020, after Maple *) CROSSREFS Columns k=0-3 give: A057427, A015723, A327605, A327628. Rows n=0,(1+2),3-5 give: A000004, A000012, A005408, A104249, A005894. Main diagonal gives: A327623. Cf. A327618, A327632. Sequence in context: A117417 A231345 A271344 * A183134 A328747 A346061 Adjacent sequences: A327619 A327620 A327621 * A327623 A327624 A327625 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Sep 19 2019 STATUS approved

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Last modified August 10 22:52 EDT 2024. Contains 375059 sequences. (Running on oeis4.)