A327618 - OEIS

%I #26 Apr 30 2020 07:24:44

%S 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,

%T 39,109,100,35,1,0,1,13,56,219,315,274,54,1,0,1,15,76,386,769,1179,

%U 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

%N 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.

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

%H Alois P. Heinz, <a href="/A327618/b327618.txt">Antidiagonals n = 0..200, flattened</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>

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

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

%e Square array A(n,k) begins:

%e   0,  0,   0,    0,     0,     0,     0,      0, ...

%e   1,  1,   1,    1,     1,     1,     1,      1, ...

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

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

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

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

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

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

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

%p      `if`(k=0, [1, 1], `if`(i<2, 0, b(n, i-1, k))+

%p          (h-> (f-> f +[0, f[1]*h[2]/h[1]])(h[1]*

%p         b(n-i, min(n-i, i), k)))(b(i$2, k-1))))

%p     end:

%p A:= (n, k)-> b(n$2, k)[2]:

%p seq(seq(A(n, d-n), n=0..d), d=0..14);

%t 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]]]];

%t A[n_, k_] := b[n, n, k][[2]];

%t Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Apr 30 2020, after _Alois P. Heinz_ *)

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

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

%Y Main diagonal gives A327619.

%Y Cf. A323718, A327622, A327631.

%K nonn,tabl

%O 0,9

%A _Alois P. Heinz_, Sep 19 2019