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 A344649 #13 Jun 09 2021 06:23:52
%S A344649 1,0,1,0,0,1,0,1,0,1,0,1,1,0,1,0,1,2,1,0,1,0,1,3,2,1,0,1,0,1,3,3,2,1,
%T A344649 0,1,0,1,4,4,3,2,1,0,1,0,1,5,6,4,3,2,1,0,1,0,1,7,7,6,4,3,2,1,0,1,0,1,
%U A344649 8,10,8,6,4,3,2,1,0,1
%N A344649 Triangle read by rows where T(n,k) is the number of strict integer partitions of 2n with reverse-alternating sum 2k.
%C A344649 The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. This is equal to (-1)^(m-1) times the number of odd parts in the conjugate partition, where m is the number of parts. So T(n,k) is the number of strict integer partitions of 2n into an odd number of parts whose conjugate has exactly 2k odd parts.
%C A344649 Also the number of reversed strict integer partitions of 2n with alternating sum 2k.
%e A344649 Triangle begins:
%e A344649 1
%e A344649 0 1
%e A344649 0 0 1
%e A344649 0 1 0 1
%e A344649 0 1 1 0 1
%e A344649 0 1 2 1 0 1
%e A344649 0 1 3 2 1 0 1
%e A344649 0 1 3 3 2 1 0 1
%e A344649 0 1 4 4 3 2 1 0 1
%e A344649 0 1 5 6 4 3 2 1 0 1
%e A344649 0 1 7 7 6 4 3 2 1 0 1
%e A344649 0 1 8 10 8 6 4 3 2 1 0 1
%e A344649 0 1 10 13 12 8 6 4 3 2 1 0 1
%e A344649 0 1 11 18 15 12 8 6 4 3 2 1 0 1
%e A344649 0 1 14 22 21 16 12 8 6 4 3 2 1 0 1
%e A344649 0 1 15 29 27 23 16 12 8 6 4 3 2 1 0 1
%e A344649 Row n = 8 counts the following partitions (empty columns indicated by dots):
%e A344649 . (8,7,1) (7,6,3) (7,5,4) (9,4,3) (11,3,2) (13,2,1) . (16)
%e A344649 (8,6,2) (8,5,3) (10,4,2) (12,3,1)
%e A344649 (9,6,1) (9,5,2) (11,4,1)
%e A344649 (6,4,3,2,1) (10,5,1)
%e A344649 Row n = 9 counts the following partitions (empty columns indicated by dots, A..I = 10..18):
%e A344649 . 981 873 765 954 B43 D32 F21 . I
%e A344649 972 864 A53 C42 E31
%e A344649 A71 963 B52 D41
%e A344649 65421 A62 C51
%e A344649 75321 B61
%e A344649 84321
%t A344649 sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
%t A344649 Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&sats[#]==k&]],{n,0,30,2},{k,0,n,2}]
%Y A344649 The non-reversed version is A152146.
%Y A344649 The non-reversed non-strict version is A239830.
%Y A344649 Column k = 2 is A343941.
%Y A344649 The non-strict version is A344610.
%Y A344649 Row sums are A344650.
%Y A344649 Right half of even-indexed rows of A344739.
%Y A344649 A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
%Y A344649 A067659 counts strict partitions of odd length.
%Y A344649 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A344649 A120452 counts partitions of 2n with reverse-alternating sum 2.
%Y A344649 A124754 gives alternating sum of standard compositions (reverse: A344618).
%Y A344649 A316524 is the alternating sum of the prime indices of n (reverse: A344616).
%Y A344649 A325534/A325535 count separable/inseparable partitions.
%Y A344649 A344604 counts wiggly compositions with twins.
%Y A344649 A344611 counts partitions of 2n with reverse-alternating sum >= 0.
%Y A344649 A344741 counts partitions of 2n with reverse-alternating sum -2.
%Y A344649 Cf. A000070, A000097, A003242, A006330, A027187, A114121, A116406, A239829, A344607, A344608, A344651, A344654.
%K A344649 nonn,tabl
%O A344649 0,18
%A A344649 _Gus Wiseman_, Jun 05 2021