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 A357704 #7 Oct 10 2022 20:47:12
%S A357704 1,0,1,0,0,2,0,0,1,2,0,0,2,0,3,0,0,2,2,0,3,0,0,3,1,3,0,4,0,0,3,2,4,2,
%T A357704 0,4,0,0,4,2,6,2,3,0,5,0,0,4,3,5,7,3,3,0,5,0,0,5,3,8,4,10,2,4,0,6,0,0,
%U A357704 5,4,8,6,11,9,3,4,0,6,0,0,6,4,11,5,15,8,13,3,5,0,7
%N A357704 Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with half-alternating sum k, where k ranges from -n to n in steps of 2.
%C A357704 We define the half-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A + B - C - D + E + F - G - ...
%e A357704 Triangle begins:
%e A357704 1
%e A357704 0 1
%e A357704 0 0 2
%e A357704 0 0 1 2
%e A357704 0 0 2 0 3
%e A357704 0 0 2 2 0 3
%e A357704 0 0 3 1 3 0 4
%e A357704 0 0 3 2 4 2 0 4
%e A357704 0 0 4 2 6 2 3 0 5
%e A357704 0 0 4 3 5 7 3 3 0 5
%e A357704 0 0 5 3 8 4 10 2 4 0 6
%e A357704 0 0 5 4 8 6 11 9 3 4 0 6
%e A357704 0 0 6 4 11 5 15 8 13 3 5 0 7
%e A357704 0 0 6 5 11 8 13 19 10 13 4 5 0 7
%e A357704 0 0 7 5 14 8 19 13 25 9 17 4 6 0 8
%e A357704 0 0 7 6 14 11 19 17 29 23 13 18 5 6 0 8
%e A357704 Row n = 7 counts the following reversed partitions:
%e A357704 . . (115) (124) (133) (11113) . (7)
%e A357704 (1114) (1222) (223) (111112) (16)
%e A357704 (1123) (11122) (25)
%e A357704 (1111111) (34)
%t A357704 halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]),{i,Length[f]}];
%t A357704 Table[Length[Select[Reverse/@IntegerPartitions[n],halfats[#]==k&]],{n,0,15},{k,-n,n,2}]
%Y A357704 Row sums are A000041.
%Y A357704 Last entry of row n is A008619(n).
%Y A357704 The central column in the non-reverse case is A035363, skew A035544.
%Y A357704 For original reverse-alternating sum we have A344612.
%Y A357704 For original alternating sum we have A344651, ordered A097805.
%Y A357704 The non-reverse version is A357637, skew A357638.
%Y A357704 The central column is A357639, skew A357640.
%Y A357704 The non-reverse ordered version (compositions) is A357645, skew A357646.
%Y A357704 The skew-alternating version is A357705.
%Y A357704 A351005 = alternately equal and unequal partitions, compositions A357643.
%Y A357704 A351006 = alternately unequal and equal partitions, compositions A357644.
%Y A357704 A357621 gives half-alternating sum of standard compositions, skew A357623.
%Y A357704 A357629 gives half-alternating sum of prime indices, skew A357630.
%Y A357704 A357633 gives half-alternating sum of Heinz partition, skew A357634.
%Y A357704 Cf. A029862, A053251, A357136, A357189, A357487, A357488, A357631, A357632, A357641.
%K A357704 nonn,tabl
%O A357704 0,6
%A A357704 _Gus Wiseman_, Oct 10 2022