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A380125 Total number of ways of partitioning n and any natural number less than or equal to n into the same number of parts, treating partitions of n and itself in a different order as distinct.

%I A380125 #26 Aug 24 2025 16:05:44
%S A380125 1,1,3,6,15,28,65,119,244,450,851,1504,2760,4732,8266,13958,23642,
%T A380125 38886,64339,103755,167785,266295,422014,658875,1027992,1581983,
%U A380125 2429719,3692762,5595987,8401561,12581456,18682756,27664577,40675705,59616335,86831979,126099127,182065162
%N A380125 Total number of ways of partitioning n and any natural number less than or equal to n into the same number of parts, treating partitions of n and itself in a different order as distinct.
%H A380125 Alois P. Heinz, <a href="/A380125/b380125.txt">Table of n, a(n) for n = 0..1000</a>
%F A380125 a(n) = Sum_{i=0..n} Sum_{j=0..i} p(n,j)*p(i,j), where p(n,j) is the number of partitions of n into j positive parts (A008284, A072233).
%F A380125 a(n) = A380124(n) + A238312(n).
%e A380125 For example, a(4)=15:
%e A380125   4 and 1: (4,1),
%e A380125   4 and 2: (4,2) (3+1,1+1) (2+2,1+1),
%e A380125   4 and 3: (4,3) (3+1,2+1) (2+2,2+1) (2+1+1,1+1+1),
%e A380125   4 and 4: (4,4) (3+1,3+1) (3+1,2+2) (2+2,3+1) (2+2,2+2) (2+1+1,2+1+1) (1+1+1+1,1+1+1+1).
%e A380125 Note that (3+1,2+2) and (2+2,3+1) are both counted.
%Y A380125 Similar to A380124, A380126.
%Y A380125 Cf. A008284, A072233, A238312.
%K A380125 nonn,changed
%O A380125 0,3
%A A380125 _Aidan Markey_, Jan 12 2025
%E A380125 More terms from _Chai Wah Wu_, Feb 19 2025
%E A380125 a(0)=1 prepended by _Alois P. Heinz_, Aug 24 2025