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 A174713 #24 Dec 26 2023 12:10:22
%S A174713 1,1,1,1,2,1,2,1,2,3,2,2,4,2,2,3,5,3,4,3,6,4,4,3,5,8,5,6,6,5,10,6,8,6,
%T A174713 5,7,12,8,10,9,10,7,15,10,12,12,10,7,11,18,12,16,15,15,14,11,22,15,20,
%U A174713 18,20,14,11,15
%N A174713 Triangle read by rows, A173305 (A000009 shifted down twice) * A174712 (diagonalized variant of A000041).
%C A174713 Row sums = A000041, the partition numbers.
%C A174713 The current triangle is the 2nd in an infinite set, followed by A174714 (k=3), and A174715, (k=4); in which row sums of each triangle = A000041.
%C A174713 k-th triangle in the infinite set can be defined as having the sequence:
%C A174713 "Euler transform of ones: (1,1,1,...) interleaved with (k-1) zeros"; shifted down k times (except column 0) in successive columns, then multiplied * triangle A174712, the diagonalized variant of A000041, A174713 begins with A000009 shifted down twice (triangle A173305); where A000009 = the Euler transform of period 2 sequence: [1,0,1,0,...].
%C A174713 Similarly, triangle A174714 begins with A000716 shifted down thrice; where A000716 = the Euler transform of period 3 series: [1,1,0,1,1,0,...]. Then multiply the latter as an infinite lower triangular matrix * A174712, the diagonalized variant of A000041, obtaining triangle A174714 with row sums = A000041.
%C A174713 Case k=4 = triangle A174715 which begins with the Euler transform of period 4 series: [1,1,1,0,1,1,1,0,...], shifted down 4 times in successive columns then multiplied * A174712, the diagonalized variant of A000041.
%C A174713 All triangles in the infinite set have row sums = A000041.
%C A174713 The sequences: "Euler transform of ones interleaved with (k-1) zeros" have the following properties, beginning with k=2:
%C A174713 ...
%C A174713 k=2, A000009: = Euler transform of [1,0,1,0,1,0,...] and satisfies
%C A174713 .....A000009. = p(x)/p(x^2), where p(x) = polcoeff A000041; and A000041 =
%C A174713 .....A000009(x) = r(x), then p(x) = r(x) * r(x^2) * r(x^4) * r(x^8) * ...
%C A174713 ...
%C A174713 k=3, A000726: = Euler transform of [1,1,0,1,1,0,...] and satisfies
%C A174713 .....A000726(x): = p(x)/p(x^3), and given s(x) = polcoeff A000726, we get
%C A174713 .....A000041(x) = p(x) = s(x) * s(x^3) * s(x^9) * s(x^27) * ...
%C A174713 ...
%C A174713 k=4, A001935: = Euler transform of [1,1,1,0,1,1,1,0,...] and satisfies
%C A174713 .....A001935(x) = p(x)/p(x^4) and given t(x) = polcoeff A001935, we get
%C A174713 .....A000041(x) = p(x) = t(x) * t(x^4) * t(x^16) * t(x^64) * ...
%C A174713 ...
%C A174713 Also the number of integer partitions of n whose even parts sum to k, for k an even number from zero to n. The version including odd k is A113686. - _Gus Wiseman_, Oct 23 2023
%F A174713 As infinite lower triangular matrices, A173305 * A174712.
%F A174713 T(n,k) = A000009(n-2k) * A000041(k). - _Gus Wiseman_, Oct 23 2023
%e A174713 First few rows of the triangle =
%e A174713 1;
%e A174713 1;
%e A174713 1, 1;
%e A174713 2, 1;
%e A174713 2, 1, 2;
%e A174713 3, 2, 2;
%e A174713 4, 2, 2, 3;
%e A174713 5, 3, 4, 3;
%e A174713 6, 4, 4, 3, 5;
%e A174713 8, 5, 6, 6, 5;
%e A174713 10, 6, 8, 6, 5, 7;
%e A174713 12, 8, 10, 9, 10, 7;
%e A174713 15, 10, 12, 12, 10, 7, 11;
%e A174713 18, 12, 16, 15, 15, 14, 11;
%e A174713 22, 15, 20, 18, 20, 14, 11, 15;
%e A174713 ...
%e A174713 From _Gus Wiseman_, Oct 23 2023: (Start)
%e A174713 Row n = 9 counts the following partitions:
%e A174713 (9) (72) (54) (63) (81)
%e A174713 (711) (5211) (522) (6111) (621)
%e A174713 (531) (3321) (4311) (432) (441)
%e A174713 (51111) (321111) (411111) (42111) (4221)
%e A174713 (333) (21111111) (32211) (3222) (22221)
%e A174713 (33111) (2211111) (222111)
%e A174713 (3111111)
%e A174713 (111111111)
%e A174713 (End)
%t A174713 Table[Length[Select[IntegerPartitions[n],Total[Select[#,EvenQ]]==k&]],{n,0,15},{k,0,n,2}] (* _Gus Wiseman_, Oct 23 2023 *)
%Y A174713 Cf. A000009, A173305, A174712, A174714, A174715.
%Y A174713 Row sums are A000041.
%Y A174713 The odd version is A365067.
%Y A174713 The corresponding rank statistic is A366531, odd version A366528.
%Y A174713 A000009 counts partitions into odd parts, ranks A066208.
%Y A174713 A113685 counts partitions by sum of odd parts, even version A113686.
%Y A174713 A239261 counts partitions with (sum of odd parts) = (sum of even parts).
%Y A174713 Cf. A035363, A066967, A130780, A171966, A182616, A366527, A366533.
%K A174713 nonn,tabf
%O A174713 0,5
%A A174713 _Gary W. Adamson_, Mar 27 2010