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 A344651 #12 Dec 10 2021 11:11:12
%S A344651 1,1,1,1,2,1,2,2,1,4,2,1,3,5,2,1,7,5,2,1,5,9,5,2,1,12,10,5,2,1,7,17,
%T A344651 10,5,2,1,19,19,10,5,2,1,11,28,20,10,5,2,1,30,33,20,10,5,2,1,15,47,35,
%U A344651 20,10,5,2,1,45,57,36,20,10,5,2,1,22,73,62,36,20,10,5,2,1
%N A344651 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with alternating sum k, with k ranging from n mod 2 to n in steps of 2.
%C A344651 The alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. This is equal to the number of odd parts in the conjugate partition, so T(n,k) is the number of integer partitions of n with k odd parts in the conjugate partition, which is also the number of partitions of n with k odd parts.
%C A344651 Also the number of integer partitions of n with odd-indexed parts (odd bisection) summing to k, ceiling(n/2) <= k <= n. The even-indexed version is A346633. - _Gus Wiseman_, Nov 29 2021
%e A344651 Triangle begins:
%e A344651 1
%e A344651 1
%e A344651 1 1
%e A344651 2 1
%e A344651 2 2 1
%e A344651 4 2 1
%e A344651 3 5 2 1
%e A344651 7 5 2 1
%e A344651 5 9 5 2 1
%e A344651 12 10 5 2 1
%e A344651 7 17 10 5 2 1
%e A344651 19 19 10 5 2 1
%e A344651 11 28 20 10 5 2 1
%e A344651 30 33 20 10 5 2 1
%e A344651 15 47 35 20 10 5 2 1
%e A344651 45 57 36 20 10 5 2 1
%e A344651 22 73 62 36 20 10 5 2 1
%e A344651 67 92 64 36 20 10 5 2 1
%e A344651 30 114 102 65 36 20 10 5 2 1
%e A344651 97 147 107 65 36 20 10 5 2 1
%e A344651 Row n = 10 counts the following partitions (A = 10):
%e A344651 (55) (64) (73) (82) (91) (A)
%e A344651 (3322) (442) (433) (622) (811)
%e A344651 (4411) (541) (532) (721)
%e A344651 (222211) (3331) (631) (7111)
%e A344651 (331111) (4222) (5221) (61111)
%e A344651 (22111111) (4321) (6211)
%e A344651 (1111111111) (5311) (42211)
%e A344651 (22222) (52111)
%e A344651 (32221) (511111)
%e A344651 (33211) (4111111)
%e A344651 (43111)
%e A344651 (322111)
%e A344651 (421111)
%e A344651 (2221111)
%e A344651 (3211111)
%e A344651 (31111111)
%e A344651 (211111111)
%e A344651 The conjugate version is:
%e A344651 (A) (55) (3331) (331111) (31111111) (1111111111)
%e A344651 (64) (73) (5311) (511111) (211111111)
%e A344651 (82) (91) (7111) (3211111)
%e A344651 (442) (433) (33211) (4111111)
%e A344651 (622) (532) (43111) (22111111)
%e A344651 (4222) (541) (52111)
%e A344651 (22222) (631) (61111)
%e A344651 (721) (322111)
%e A344651 (811) (421111)
%e A344651 (3322) (2221111)
%e A344651 (4321)
%e A344651 (4411)
%e A344651 (5221)
%e A344651 (6211)
%e A344651 (32221)
%e A344651 (42211)
%e A344651 (222211)
%t A344651 ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
%t A344651 Table[Length[Select[IntegerPartitions[n],ats[#]==k&]],{n,0,15},{k,Mod[n,2],n,2}]
%Y A344651 This is A103919 with all zeros removed.
%Y A344651 The strict version is A152146 interleaved with A152157.
%Y A344651 The rows are those of A239830 interleaved with those of A239829.
%Y A344651 The reverse version is the right half of A344612.
%Y A344651 The strict reverse version is the right half of A344739.
%Y A344651 A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
%Y A344651 A027187 counts partitions with rev-alternating sum <= 0, ranked by A028260.
%Y A344651 A124754 lists alternating sums of standard compositions (reverse: A344618).
%Y A344651 A316524 is the alternating sum of the prime indices of n (reverse: A344616).
%Y A344651 A325534/A325535 count separable/inseparable partitions.
%Y A344651 A344607 counts partitions with rev-alternating sum >= 0, ranked by A344609.
%Y A344651 A344608 counts partitions with rev-alternating sum < 0, ranked by A119899.
%Y A344651 A344610 counts partitions of n by positive rev-alternating sum.
%Y A344651 A344611 counts partitions of 2n with rev-alternating sum >= 0.
%Y A344651 A345197 counts compositions by sum, length, and alternating sum.
%Y A344651 A346697 gives the sum of odd-indexed prime indices (reverse: A346699).
%Y A344651 A346702 represents the odd bisection of compositions, sums A209281.
%Y A344651 Cf. A000070, A000097, A003242, A006330, A025047, A097805, A114121, A116406, A131577, A344617, A344649, A344650, A344654, A346633.
%K A344651 nonn,tabf
%O A344651 0,5
%A A344651 _Gus Wiseman_, Jun 05 2021