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A232623 Number of partitions of 2*n into parts with multiplicity <= n.

%I A232623 #27 Jul 17 2025 09:41:13
%S A232623 1,1,4,9,19,37,70,124,216,363,597,960,1519,2359,3617,5469,8173,12079,
%T A232623 17680,25630,36848,52547,74383,104556,146018,202651,279631,383719,
%U A232623 523813,711502,961902,1294552,1734788,2315171,3077592,4075658,5377900,7071523,9267454
%N A232623 Number of partitions of 2*n into parts with multiplicity <= n.
%H A232623 Paolo Xausa, <a href="/A232623/b232623.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)
%F A232623 a(n) = A061199(n,2*n).
%F A232623 a(n) ~ exp(2*Pi*sqrt(n/3))/(8*n*sqrt(3)). - _Vaclav Kotesovec_, Nov 27 2013
%F A232623 a(n) = A000041(2*n) - A000041(n-1). - _Alan Michael Gómez Calderón_, Jul 12 2025
%e A232623 a(1) = 1: [2].
%e A232623 a(2) = 4: [2,1,1], [2,2], [3,1], [4].
%e A232623 a(3) = 9: [2,2,1,1], [2,2,2], [3,1,1,1], [3,2,1], [3,3], [4,1,1], [4,2], [5,1], [6].
%e A232623 a(4) = 19: [2,2,1,1,1,1], [2,2,2,1,1], [2,2,2,2], [3,2,1,1,1], [3,2,2,1], [3,3,1,1], [3,3,2], [4,1,1,1,1], [4,2,1,1], [4,2,2], [4,3,1], [4,4], [5,1,1,1], [5,2,1], [5,3], [6,1,1], [6,2], [7,1], [8].
%p A232623 b:= proc(n, i, k) option remember; `if`(n=0, 1,
%p A232623       `if`(i>n, 0, add(b(n-i*j, i+1, min(k,
%p A232623        iquo(n-i*j, i+1))), j=0..min(n/i, k))))
%p A232623     end:
%p A232623 a:= n-> b(2*n, 1, n):
%p A232623 seq(a(n), n=0..50);
%t A232623 b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i>n, 0, Sum[b[n-i*j, i+1, Min[k, Quotient[n-i*j, i+1]]], {j, 0, Min[n/i, k]}]]]; a[n_] := b[2*n, 1, n]; Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Feb 11 2015, after _Alois P. Heinz_ *)
%t A232623 A232623[n_] := PartitionsP[2*n] - PartitionsP[n - 1];
%t A232623 Array[A232623, 50, 0] (* _Paolo Xausa_, Jul 17 2025 *)
%o A232623 (PARI) a(n) = numbpart(2*n) - numbpart(n-1); \\ _Michel Marcus_, Jul 13 2025
%Y A232623 Cf. A000041, A061199, A232605, A232697.
%K A232623 nonn
%O A232623 0,3
%A A232623 _Alois P. Heinz_, Nov 27 2013