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A327618 Number A(n,k) of parts in all k-times partitions of n; square array A(n,k), n>=0, k>=0, read by antidiagonals. 7
0, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 5, 6, 1, 0, 1, 7, 14, 12, 1, 0, 1, 9, 25, 44, 20, 1, 0, 1, 11, 39, 109, 100, 35, 1, 0, 1, 13, 56, 219, 315, 274, 54, 1, 0, 1, 15, 76, 386, 769, 1179, 581, 86, 1, 0, 1, 17, 99, 622, 1596, 3643, 3234, 1417, 128, 1, 0, 1, 19, 125, 939, 2960, 9135, 12336, 10789, 2978, 192, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

Row n is binomial transform of the n-th row of triangle A327631.

LINKS

Alois P. Heinz, Antidiagonals n = 0..200, flattened

Wikipedia, Partition (number theory)

FORMULA

A(n,k) = Sum_{i=0..k} binomial(k,i) * A327631(n,i).

EXAMPLE

A(2,2) = 5 = 1+2+2 counting the parts in 2, 11, 1|1.

Square array A(n,k) begins:

0, 0, 0, 0, 0, 0, 0, 0, ...

1, 1, 1, 1, 1, 1, 1, 1, ...

1, 3, 5, 7, 9, 11, 13, 15, ...

1, 6, 14, 25, 39, 56, 76, 99, ...

1, 12, 44, 109, 219, 386, 622, 939, ...

1, 20, 100, 315, 769, 1596, 2960, 5055, ...

1, 35, 274, 1179, 3643, 9135, 19844, 38823, ...

1, 54, 581, 3234, 12336, 36911, 93302, 208377, ...

MAPLE

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0],

`if`(k=0, [1, 1], `if`(i<2, 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), 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] = If[n == 0, {1, 0}, If[k == 0, {1, 1}, If[i < 2, 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], 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, Apr 30 2020, after Alois P. Heinz *)

CROSSREFS

Columns k=0-3 give: A057427, A006128, A327594, A327627.

Rows n=0-3 give: A000004, A000012, A005408, A095794(k+1).

Main diagonal gives A327619.

Cf. A323718, A327622, A327631.

Sequence in context: A285574 A354821 A081719 * A121314 A119271 A323222

Adjacent sequences: A327615 A327616 A327617 * A327619 A327620 A327621

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Sep 19 2019

STATUS

approved

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Last modified December 2 20:59 EST 2022. Contains 358510 sequences. (Running on oeis4.)