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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
OFFSET
0,14
COMMENTS
Row n is binomial transform of the n-th row of triangle A327632.
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);
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.
Sequence in context: A117417 A231345 A271344 * A183134 A328747 A346061
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 19 2019
STATUS
approved