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 A380959 #11 Feb 11 2025 00:00:02
%S A380959 1,1,0,1,0,0,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,0,1,1,1,2,0,0,
%T A380959 0,0,1,1,1,2,0,0,0,0,0,1,1,1,2,1,0,0,0,0,0,1,1,1,2,1,1,0,0,0,0,0,1,1,
%U A380959 1,2,1,2,0,0,0,0,0,0
%N A380959 Array read by antidiagonals downward where A(n,k) is the number of integer partitions of k with product n.
%C A380959 Counts finite multisets of positive integers by product and sum.
%F A380959 A(n,k) = A379666(k,n).
%e A380959 Array begins:
%e A380959 k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9 k10 k11 k12
%e A380959 ---------------------------------------------------
%e A380959 n=1: 1 1 1 1 1 1 1 1 1 1 1 1 1
%e A380959 n=2: 0 0 1 1 1 1 1 1 1 1 1 1 1
%e A380959 n=3: 0 0 0 1 1 1 1 1 1 1 1 1 1
%e A380959 n=4: 0 0 0 0 2 2 2 2 2 2 2 2 2
%e A380959 n=5: 0 0 0 0 0 1 1 1 1 1 1 1 1
%e A380959 n=6: 0 0 0 0 0 1 2 2 2 2 2 2 2
%e A380959 n=7: 0 0 0 0 0 0 0 1 1 1 1 1 1
%e A380959 n=8: 0 0 0 0 0 0 2 2 3 3 3 3 3
%e A380959 n=9: 0 0 0 0 0 0 1 1 1 2 2 2 2
%e A380959 n=10: 0 0 0 0 0 0 0 1 1 1 2 2 2
%e A380959 n=11: 0 0 0 0 0 0 0 0 0 0 0 1 1
%e A380959 n=12: 0 0 0 0 0 0 0 2 3 3 3 3 4
%e A380959 The A(12,9) = 3 partitions are: (6,2,1), (4,3,1,1), (3,2,2,1,1).
%e A380959 The A(9,12) = 2 partitions are: (9,1,1,1), (3,3,1,1,1,1,1,1).
%t A380959 nn=12;
%t A380959 tt=Table[Length[Select[IntegerPartitions[k],Times@@#==n&]],{n,1,nn},{k,0,nn}] (* array *)
%t A380959 tr=Table[tt[[j,i-j]],{i,2,nn},{j,i-1}] (* antidiagonals *)
%t A380959 Join@@tr (* sequence *)
%Y A380959 Column sums are A000041 = partitions of n, strict A000009, no ones A002865.
%Y A380959 Diagonal A(n,n) is A001055(n) = factorizations of n, strict A045778.
%Y A380959 Row n converges to A001055(n).
%Y A380959 Lower triangle is A319000.
%Y A380959 Transpose of A379666.
%Y A380959 Antidiagonal sums are A379667, without ones A379669 (zeros A379670), strict A379672.
%Y A380959 A316439 counts factorizations by length, partitions A008284.
%Y A380959 A326622 counts factorizations with integer mean, strict A328966.
%Y A380959 Counting and ranking multisets by comparing sum and product:
%Y A380959 - same: A001055, ranks A301987
%Y A380959 - divisible: A057567, ranks A326155
%Y A380959 - divisor: A057568, ranks A326149, see A379733
%Y A380959 - greater than: A096276 shifted right, ranks A325038
%Y A380959 - greater or equal: A096276, ranks A325044
%Y A380959 - less than: A114324, ranks A325037, see A318029
%Y A380959 - less or equal: A319005, ranks A379721, see A025147
%Y A380959 - different: A379736, ranks A379722, see A111133
%Y A380959 Cf. A003963, A028422, A069016, A319916, A319057, A325036, A325041, A325042, A326152.
%Y A380959 Cf. A318950, A379668, A379671, A379678.
%K A380959 nonn,tabl
%O A380959 0,32
%A A380959 _Gus Wiseman_, Feb 10 2025