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 A344612 #14 Jan 06 2024 14:32:07
%S A344612 1,0,1,0,1,1,0,1,1,1,0,1,2,1,1,0,1,2,2,1,1,0,1,2,3,3,1,1,0,1,2,4,3,3,
%T A344612 1,1,0,1,2,4,5,5,3,1,1,0,1,2,4,7,5,6,3,1,1,0,1,2,4,8,7,9,6,3,1,1,0,1,
%U A344612 2,4,8,12,7,11,6,3,1,1,0,1,2,4,8,14,11,14,12,6,3,1,1
%N A344612 Triangle read by rows where T(n,k) is the number of integer partitions of n with reverse-alternating sum k ranging from -n to n in steps of 2.
%C A344612 The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. This is also (-1)^(k-1) times the sum of the even-indexed parts minus the sum of the odd-indexed parts.
%C A344612 Also the number of reversed integer partitions of n with alternating sum k ranging from -n to n in steps of 2.
%C A344612 Also the number of integer partitions of n with (-1)^(m-1) * b = k where m is the greatest part and b is the number of odd parts, with k ranging from -n to n in steps of 2.
%H A344612 Andrew Howroyd, <a href="/A344612/b344612.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%e A344612 Triangle begins:
%e A344612 1
%e A344612 0 1
%e A344612 0 1 1
%e A344612 0 1 1 1
%e A344612 0 1 2 1 1
%e A344612 0 1 2 2 1 1
%e A344612 0 1 2 3 3 1 1
%e A344612 0 1 2 4 3 3 1 1
%e A344612 0 1 2 4 5 5 3 1 1
%e A344612 0 1 2 4 7 5 6 3 1 1
%e A344612 0 1 2 4 8 7 9 6 3 1 1
%e A344612 0 1 2 4 8 12 7 11 6 3 1 1
%e A344612 0 1 2 4 8 14 11 14 12 6 3 1 1
%e A344612 0 1 2 4 8 15 19 11 18 12 6 3 1 1
%e A344612 0 1 2 4 8 15 24 15 23 20 12 6 3 1 1
%e A344612 0 1 2 4 8 15 26 30 15 31 21 12 6 3 1 1
%e A344612 For example, row n = 7 counts the following partitions:
%e A344612 (61) (52) (43) (331) (322) (511) (7)
%e A344612 (4111) (2221) (22111) (421)
%e A344612 (3211) (1111111) (31111)
%e A344612 (211111)
%e A344612 Row n = 9 counts the following partitions:
%e A344612 81 72 63 54 441 333 522 711 9
%e A344612 6111 4221 3222 22221 432 621
%e A344612 5211 3321 33111 531 51111
%e A344612 411111 4311 2211111 32211
%e A344612 222111 111111111 42111
%e A344612 321111 3111111
%e A344612 21111111
%t A344612 sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
%t A344612 Table[Length[Select[IntegerPartitions[n],sats[#]==k&]],{n,0,15},{k,-n,n,2}]
%o A344612 (PARI) row(n)={my(v=vector(n+1)); forpart(p=n, my(s=-sum(i=1, #p, p[i]*(-1)^i)); v[(s+n)/2+1]++); v} \\ _Andrew Howroyd_, Jan 06 2024
%Y A344612 Row sums are A000041.
%Y A344612 The midline k = n/2 is also A000041.
%Y A344612 The right half (i.e., k >= 0) for even n is A344610.
%Y A344612 The rows appear to converge to A344611 (from left) and A006330 (from right).
%Y A344612 The non-reversed version is A344651 (A239830 interleaved with A239829).
%Y A344612 The strict version is A344739.
%Y A344612 A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
%Y A344612 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A344612 A120452 counts partitions of 2n with rev-alt sum 2 (negative: A344741).
%Y A344612 A316524 is the alternating sum of the prime indices of n (reverse: A344616).
%Y A344612 A325534/A325535 count separable/inseparable partitions.
%Y A344612 A344618 gives reverse-alternating sums of standard compositions.
%Y A344612 Cf. A000070, A000097, A003242, A027187, A124754, A152146, A344607, A344608, A344649, A344650, A344654.
%K A344612 nonn,tabl
%O A344612 0,13
%A A344612 _Gus Wiseman_, Jun 01 2021