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A357637 Triangle read by rows where T(n,k) is the number of integer partitions of n with half-alternating sum k, where k ranges from -n to n in steps of 2.

%I A357637 #16 Oct 12 2022 14:19:53
%S A357637 1,0,1,0,0,2,0,0,1,2,0,0,1,1,3,0,0,0,2,2,3,0,0,0,0,5,2,4,0,0,0,0,2,6,
%T A357637 3,4,0,0,0,0,2,3,9,3,5,0,0,0,0,0,4,7,10,4,5,0,0,0,0,0,0,11,8,13,4,6,0,
%U A357637 0,0,0,0,0,4,15,12,14,5,6,0,0,0,0,0,0,3,7,25,13,17,5,7
%N A357637 Triangle read by rows where T(n,k) is the number of integer partitions of n with half-alternating sum k, where k ranges from -n to n in steps of 2.
%C A357637 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 - ...
%H A357637 Alois P. Heinz, <a href="/A357637/b357637.txt">Rows n = 0..200, flattened</a>
%F A357637 Conjecture: The column sums are A029862.
%e A357637 Triangle begins:
%e A357637   1
%e A357637   0  1
%e A357637   0  0  2
%e A357637   0  0  1  2
%e A357637   0  0  1  1  3
%e A357637   0  0  0  2  2  3
%e A357637   0  0  0  0  5  2  4
%e A357637   0  0  0  0  2  6  3  4
%e A357637   0  0  0  0  2  3  9  3  5
%e A357637   0  0  0  0  0  4  7 10  4  5
%e A357637   0  0  0  0  0  0 11  8 13  4  6
%e A357637   0  0  0  0  0  0  4 15 12 14  5  6
%e A357637   0  0  0  0  0  0  3  7 25 13 17  5  7
%e A357637 Row n = 9 counts the following partitions:
%e A357637   (3222)       (333)      (432)     (441)  (9)
%e A357637   (22221)      (3321)     (522)     (531)  (54)
%e A357637   (21111111)   (4221)     (4311)    (621)  (63)
%e A357637   (111111111)  (32211)    (5211)    (711)  (72)
%e A357637                (222111)   (6111)           (81)
%e A357637                (2211111)  (33111)
%e A357637                (3111111)  (42111)
%e A357637                           (51111)
%e A357637                           (321111)
%e A357637                           (411111)
%p A357637 b:= proc(n, i, s, t) option remember; `if`(n=0, x^s, `if`(i<1, 0,
%p A357637       b(n, i-1, s, t)+b(n-i, min(n-i, i), s+`if`(t<2, i, -i), irem(t+1, 4))))
%p A357637     end:
%p A357637 T:= n-> (p-> seq(coeff(p, x, i), i=-n..n, 2))(b(n$2, 0$2)):
%p A357637 seq(T(n), n=0..15);  # _Alois P. Heinz_, Oct 12 2022
%t A357637 halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]),{i,Length[f]}];
%t A357637 Table[Length[Select[IntegerPartitions[n],halfats[#]==k&]],{n,0,12},{k,-n,n,2}]
%Y A357637 Row sums are A000041.
%Y A357637 Number of nonzero entries in row n appears to be A004525(n+1).
%Y A357637 Last entry of row n is A008619(n).
%Y A357637 Column sums appear to be A029862.
%Y A357637 The central column is A035363, skew A035544.
%Y A357637 For original alternating sum we have A344651, ordered A097805.
%Y A357637 The skew-alternating version is A357638.
%Y A357637 The central column of the reverse is A357639, skew A357640.
%Y A357637 The ordered version (compositions) is A357645, skew A357646.
%Y A357637 The reverse version is A357704, skew A357705.
%Y A357637 A351005 = alternately equal and unequal partitions, compositions A357643.
%Y A357637 A351006 = alternately unequal and equal partitions, compositions A357644.
%Y A357637 A357621 gives  half-alternating sum of standard compositions, skew A357623.
%Y A357637 A357629 gives half-alternating sum of prime indices, skew A357630.
%Y A357637 A357633 gives half-alternating sum of Heinz partition, skew  A357634.
%Y A357637 Cf. A053251, A357136, A357189, A357487, A357488, A357631, A357632, A357641.
%K A357637 nonn,tabl
%O A357637 0,6
%A A357637 _Gus Wiseman_, Oct 10 2022