cp's OEIS Frontend

A360072 Number of pairs of positive integers (k,i) such that k >= i and there exists an integer partition of n of length k with i distinct parts.

%I A360072 #12 Jan 31 2023 14:19:03
%S A360072 0,1,2,3,5,5,9,9,13,14,18,19,26,25,30,34,39,40,48,48,56,59,64,67,78,
%T A360072 78,84,89,97,99,111,111,121,125,131,137,149,149,158,165,176,177,190,
%U A360072 191,202,210,216,222,238,239,250,256,266,270,284,289,302,307,316,323
%N A360072 Number of pairs of positive integers (k,i) such that k >= i and there exists an integer partition of n of length k with i distinct parts.
%C A360072 This is the number of nonzero terms in the n-th triangle of A360071.
%H A360072 Andrew Howroyd, <a href="/A360072/b360072.txt">Table of n, a(n) for n = 0..1000</a>
%F A360072 a(n) = A000005(n) + Sum_{k=2..floor((sqrt(8*n+1)-1)/2)} (1 + n - binomial(k+1,2)) for n > 0. - _Andrew Howroyd_, Jan 30 2023
%e A360072 The a(5) = 5 pairs are: (1,1), (2,2), (3,2), (4,2), (5,1). The pair (3,3) is absent because it is not possible to partition 5 into 3 parts, all 3 of which are distinct.
%e A360072 The a(6) = 9 pairs are: (1,1), (2,1), (2,2), (3,1), (3,2), (3,3), (4,2), (5,2), (6,1). The pair (3,3) is present because (3,2,1) is a partition of 6 into 3 parts, all 3 of which are distinct.
%t A360072 Table[Count[Flatten[Sign[Table[Length[Select[IntegerPartitions[n], Length[#]==k&&Length[Union[#]]==i&]],{k,1,n},{i,1,k}]]],1],{n,0,30}]
%o A360072 (PARI) a(n) = if(n < 1, 0, numdiv(n) + sum(k=2, (sqrtint(8*n+1)-1)\2, n-binomial(k+1,2)+1)) \\ _Andrew Howroyd_, Jan 30 2023
%Y A360072 A000041 counts integer partitions, strict A000009.
%Y A360072 A008284 counts partitions by number of parts, reverse A058398.
%Y A360072 A116608 counts partitions by number of distinct parts.
%Y A360072 Cf. A000005, A051731, A055884, A060016, A331195, A360010, A360071.
%K A360072 nonn
%O A360072 0,3
%A A360072 _Gus Wiseman_, Jan 28 2023
%E A360072 Terms a(31) and beyond from _Andrew Howroyd_, Jan 30 2023