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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

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<n, 0, b(n, i-1, k)+

         (h-> (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);

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 A053382

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 May 28 12:18 EDT 2020. Contains 334681 sequences. (Running on oeis4.)