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 A344739 #8 Jun 08 2021 06:26:51
%S A344739 1,0,1,0,0,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,1,1,1,0,1,
%T A344739 0,1,0,1,1,1,0,1,1,0,1,0,1,1,1,1,0,2,1,0,1,0,1,1,1,2,0,1,2,1,0,1,0,1,
%U A344739 1,1,2,1,0,2,2,1,0,1
%N A344739 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with reverse-alternating sum k, with k ranging from -n to n in steps of 2.
%C A344739 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 n such that k is equal to (-1)^(m-1) times the number of odd conjugate parts.
%C A344739 By conjugation, T(n,k) is also equal to the number of integer partitions of n covering an initial interval of positive integers such that k is equal to (-1)^(r-1) times the number of odd parts, where r is the greatest part.
%C A344739 Also the number of reversed strict integer partitions of n with alternating sum k.
%e A344739 Triangle begins:
%e A344739 1
%e A344739 0 1
%e A344739 0 0 1
%e A344739 0 1 0 1
%e A344739 0 1 0 0 1
%e A344739 0 1 1 0 0 1
%e A344739 0 1 1 0 1 0 1
%e A344739 0 1 1 1 0 1 0 1
%e A344739 0 1 1 1 0 1 1 0 1
%e A344739 0 1 1 1 1 0 2 1 0 1
%e A344739 0 1 1 1 2 0 1 2 1 0 1
%e A344739 0 1 1 1 2 1 0 2 2 1 0 1
%e A344739 0 1 1 1 2 2 0 1 3 2 1 0 1
%e A344739 0 1 1 1 2 3 1 0 2 3 2 1 0 1
%e A344739 0 1 1 1 2 3 3 0 1 3 3 2 1 0 1
%e A344739 0 1 1 1 2 3 4 1 0 3 4 3 2 1 0 1
%e A344739 0 1 1 1 2 3 5 3 0 1 4 4 3 2 1 0 1
%e A344739 0 1 1 1 2 3 5 5 1 0 3 5 4 3 2 1 0 1
%e A344739 0 1 1 1 2 3 5 6 4 0 1 5 6 4 3 2 1 0 1
%e A344739 For example, the partitions counted by row n = 15 are (empty columns shown as dots, A...F = 10..15):
%e A344739 . E1 D2 C3 B4 A5 96 87 . 762 654 843 A32 C21 . F
%e A344739 9321 7431 6432 861 753 942 B31
%e A344739 8421 6531 54321 852 A41
%e A344739 7521 951
%t A344739 sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
%t A344739 Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&sats[#]==k&]],{n,0,12},{k,-n,n,2}]
%Y A344739 Row sums are A000009.
%Y A344739 The non-reverse version is A152146 interleaved with A152157.
%Y A344739 The non-strict version is A344612.
%Y A344739 The right halves of even-indexed rows are A344649.
%Y A344739 The non-reverse non-strict version is the right half of A344651, which is A239830 interleaved with A239829.
%Y A344739 A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
%Y A344739 A124754 lists alternating sums of standard compositions (reverse: A344618).
%Y A344739 A316524 is the alternating sum of the prime indices of n (reverse: A344616).
%Y A344739 A344610 counts partitions of n by positive reverse-alternating sum.
%Y A344739 A344611 counts partitions of 2n with reverse-alternating sum >= 0.
%Y A344739 Cf. A000070, A003242, A006330, A027187, A103919, A114121, A343941, A344607, A344608, A344650, A344654.
%K A344739 nonn,tabl
%O A344739 0,52
%A A344739 _Gus Wiseman_, Jun 05 2021