A384926 Number of partitions of n with six designated summands.

1, 3, 9, 22, 51, 108, 221, 414, 765, 1344, 2310, 3834, 6248, 9894, 15408, 23550, 35394, 52353, 76402, 109959, 156366, 219850, 305796, 421281, 574568, 777234, 1042083, 1387037, 1831362, 2402595, 3128995, 4051797, 5211639, 6668490, 8482089, 10737063, 13516615

Offset: 21

Author

Omar E. Pol, Jul 23 2025

Keywords

• nonn

Examples

			21 has only one partition with six designated summands: [6'+ 5'+ 4'+ 3'+ 2'+ 1'], so a(21) = 1.
22 has three partitions with six designated summands: [7'+ 5'+ 4'+ 3'+ 2'+ 1'], [6'+ 5'+ 4'+ 3'+ 2'+ 1'+ 1], [6'+ 5'+ 4'+ 3'+ 2'+ 1 + 1'], so a(22) = 3.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 21..4000

Crossrefs

Column k=6 of A385001.

Columns of A385001 converge to A000716.

Other columns k of A385001 are A000007 (k=0), A000203 (k=1), A002127 (k=2), A002128 (k=3), A365664 (k=4), A365665 (k=5).

Programs

Maple

b:= proc(n, i) option remember; series(`if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+add(b(n-i*j, i-1)*j*x, j=1..n/i))), x, 7)
    end:
a:= n-> coeff(b(n$2), x, 6):
seq(a(n), n=21..57);  # Alois P. Heinz, Jul 23 2025

Mathematica

nmax=60; Drop[CoefficientList[Series[1/13 * Sum[(-1)^k*(2*k + 1)*Binomial[k + 6, 12]*x^(k*(k + 1)/2), {k, 6, nmax}]/Sum[(-1)^k*(2*k + 1)*x^(k*(k + 1)/2), {k, 0, nmax}], {x, 0, nmax}], x] , 21] (* Vaclav Kotesovec, Jul 29 2025 *)

Formula

A000716(n) >= a(21+n) with equality only for n <= 6.

Sum_{k=1..n} a(k) ~ Pi^12 * n^12 / (12! * 13!). - Vaclav Kotesovec, Aug 01 2025

Extensions

More terms from Alois P. Heinz, Jul 23 2025