A213177 -id:A213177 - cp's OEIS frontend

A320379 Number of parts in all partitions of n with largest multiplicity nine.

9, 0, 10, 10, 21, 21, 44, 44, 80, 101, 152, 178, 279, 330, 476, 594, 813, 1004, 1359, 1669, 2208, 2743, 3540, 4365, 5610, 6876, 8697, 10669, 13349, 16290, 20258, 24603, 30334, 36773, 44999, 54312, 66122, 79490, 96214, 115337, 138887, 165891, 198941, 236783

Offset: 9

Alois P. Heinz, Oct 11 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 9..1000

Column k=9 of A213177.

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
      add((l->l+[0, l[1]*j])(b(n-i*j, i-1, k)), j=0..min(n/i, k))))
    end:
a:= n-> (k-> (b(n$2, k)-b(n$2, k-1))[2])(9):
seq(a(n), n=9..50);

a(n) ~ log(10) * exp(Pi*sqrt(3*n/5)) / (2 * Pi * 15^(1/4) * n^(1/4)). - Vaclav Kotesovec, Oct 25 2018

A320380 Number of parts in all partitions of n with largest multiplicity ten.

Original entry on oeis.org

10, 0, 11, 11, 23, 23, 48, 48, 87, 100, 164, 192, 290, 344, 500, 614, 847, 1038, 1412, 1728, 2286, 2812, 3650, 4491, 5758, 7045, 8924, 10912, 13668, 16647, 20691, 25104, 30952, 37444, 45853, 55282, 67291, 80824, 97860, 117188, 141132, 168446, 202003, 240312

Offset: 10

Alois P. Heinz, Oct 11 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 10..1000

Column k=10 of A213177.

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
      add((l->l+[0, l[1]*j])(b(n-i*j, i-1, k)), j=0..min(n/i, k))))
    end:
a:= n-> (k-> (b(n$2, k)-b(n$2, k-1))[2])(10):
seq(a(n), n=10..50);

a(n) ~ 3^(1/4) * log(11) * exp(2*Pi*sqrt(5*n/33)) / (2 * Pi * 55^(1/4) * n^(1/4)). - Vaclav Kotesovec, Oct 25 2018

A320381 Number of parts in all partitions of 2n with largest multiplicity n.

Original entry on oeis.org

0, 1, 5, 7, 15, 18, 38, 43, 81, 101, 164, 206, 332, 405, 613, 783, 1115, 1410, 1984, 2483, 3402, 4281, 5697, 7147, 9417, 11702, 15167, 18861, 24093, 29782, 37745, 46377, 58206, 71325, 88665, 108194, 133675, 162278, 199154, 241040, 293934, 354306, 429968, 516256

Offset: 0

Alois P. Heinz, Oct 11 2018

nonn

			a(2) = 5 = 3 + 2: [2,1,1], [2,2].

Vaclav Kotesovec, Table of n, a(n) for n = 0..788 (terms 0..400 from Alois P. Heinz)

Cf. A213177.

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
      add((l-> [0, l[1]*j]+l)(b(n-i*j, i-1, k)), j=0..min(n/i, k))))
    end:
a:= n-> (b(2*n$2, n)-b(2*n$2, n-1))[2]:
seq(a(n), n=0..45);

b[n_, i_, k_] := b[n, i, k] = If[n==0, {1, 0}, If[i<1, {0, 0}, Sum[b[n - i*j, i - 1, k] /. l_List :> {l[[1]], l[[2]] + l[[1]]*j}, {j, 0, Min[n/i, k]}]]];
a[n_] := (b[2n, 2n, n] - b[2n, 2n, n-1])[[2]];
a /@ Range[0, 45] (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz *)

a(n) = A213177(2n,n).

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A320379 Number of parts in all partitions of n with largest multiplicity nine.

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A320380 Number of parts in all partitions of n with largest multiplicity ten.

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A320381 Number of parts in all partitions of 2n with largest multiplicity n.

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