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A380126 Total number of ways of partitioning n and any natural number less than or equal to n into the same number of parts, not treating partitions of n and itself in a different order as distinct.

%I A380126 #24 Aug 24 2025 15:56:49
%S A380126 1,1,3,6,14,26,58,106,214,394,742,1314,2406,4139,7234,12250,20778,
%T A380126 34279,56805,91866,148816,236772,375899,588208,919235,1417538,2180608,
%U A380126 3320197,5038918,7577850,11363516,16899942,25056925,36892553,54136934,78951553,114783293,165922204
%N A380126 Total number of ways of partitioning n and any natural number less than or equal to n into the same number of parts, not treating partitions of n and itself in a different order as distinct.
%H A380126 Alois P. Heinz, <a href="/A380126/b380126.txt">Table of n, a(n) for n = 0..1000</a>
%F A380126 a(n) = Sum_{i=0..n-1} Sum_{j=0..i} p(n,j)*p(i,j) + Sum_{j=0..n} (p(n,j)*(p(n,j)+1))/2, where p(n,j) is the number of partitions of n into j positive parts (A008284, A072233).
%e A380126 For example, a(4)=14:
%e A380126   4 and 1: (4,1),
%e A380126   4 and 2: (4,2) (3+1,1+1) (2+2,1+1),
%e A380126   4 and 3: (4,3) (3+1,2+1) (2+2,2+1) (2+1+1,1+1+1),
%e A380126   4 and 4: (4,4) (3+1,3+1) (3+1,2+2) (2+2,2+2) (2+1+1,2+1+1) (1+1+1+1,1+1+1+1).
%e A380126 Note that (3+1,2+2) and (2+2,3+1) are not both counted.
%Y A380126 Similar to A380124, A380125.
%Y A380126 Cf. A008284, A072233.
%K A380126 nonn,changed
%O A380126 0,3
%A A380126 _Aidan Markey_, Jan 12 2025
%E A380126 More terms from _Chai Wah Wu_, Feb 19 2025
%E A380126 a(0)=1 prepended by _Alois P. Heinz_, Aug 24 2025