This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A384926 #36 Aug 01 2025 04:26:32
%S A384926 1,3,9,22,51,108,221,414,765,1344,2310,3834,6248,9894,15408,23550,
%T A384926 35394,52353,76402,109959,156366,219850,305796,421281,574568,777234,
%U A384926 1042083,1387037,1831362,2402595,3128995,4051797,5211639,6668490,8482089,10737063,13516615
%N A384926 Number of partitions of n with six designated summands.
%H A384926 Alois P. Heinz, <a href="/A384926/b384926.txt">Table of n, a(n) for n = 21..4000</a>
%F A384926 A000716(n) >= a(21+n) with equality only for n <= 6.
%F A384926 Sum_{k=1..n} a(k) ~ Pi^12 * n^12 / (12! * 13!). - _Vaclav Kotesovec_, Aug 01 2025
%e A384926 21 has only one partition with six designated summands: [6'+ 5'+ 4'+ 3'+ 2'+ 1'], so a(21) = 1.
%e A384926 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.
%p A384926 b:= proc(n, i) option remember; series(`if`(n=0, 1, `if`(i<1, 0,
%p A384926 b(n, i-1)+add(b(n-i*j, i-1)*j*x, j=1..n/i))), x, 7)
%p A384926 end:
%p A384926 a:= n-> coeff(b(n$2), x, 6):
%p A384926 seq(a(n), n=21..57); # _Alois P. Heinz_, Jul 23 2025
%t A384926 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 *)
%Y A384926 Column k=6 of A385001.
%Y A384926 Columns of A385001 converge to A000716.
%Y A384926 Other columns k of A385001 are A000007 (k=0), A000203 (k=1), A002127 (k=2), A002128 (k=3), A365664 (k=4), A365665 (k=5).
%K A384926 nonn
%O A384926 21,2
%A A384926 _Omar E. Pol_, Jul 23 2025
%E A384926 More terms from _Alois P. Heinz_, Jul 23 2025