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A357638 Triangle read by rows where T(n,k) is the number of integer partitions of n with skew-alternating sum k, where k ranges from -n to n in steps of 2.

%I A357638 #5 Oct 10 2022 20:47:17
%S A357638 1,0,1,0,1,1,0,1,1,1,0,0,3,1,1,0,0,1,4,1,1,0,0,1,4,4,1,1,0,0,0,4,5,4,
%T A357638 1,1,0,0,0,1,10,5,4,1,1,0,0,0,1,5,13,5,4,1,1,0,0,0,0,4,13,14,5,4,1,1,
%U A357638 0,0,0,0,1,13,17,14,5,4,1,1
%N A357638 Triangle read by rows where T(n,k) is the number of integer partitions of n with skew-alternating sum k, where k ranges from -n to n in steps of 2.
%C A357638 We define the skew-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A - B - C + D + E - F - G + ....
%F A357638 Conjecture: The columns are palindromes with sums A298311.
%e A357638 Triangle begins:
%e A357638   1
%e A357638   0  1
%e A357638   0  1  1
%e A357638   0  1  1  1
%e A357638   0  0  3  1  1
%e A357638   0  0  1  4  1  1
%e A357638   0  0  1  4  4  1  1
%e A357638   0  0  0  4  5  4  1  1
%e A357638   0  0  0  1 10  5  4  1  1
%e A357638   0  0  0  1  5 13  5  4  1  1
%e A357638   0  0  0  0  4 13 14  5  4  1  1
%e A357638   0  0  0  0  1 13 17 14  5  4  1  1
%e A357638   0  0  0  0  1  5 28 18 14  5  4  1  1
%e A357638 Row n = 7 counts the following partitions:
%e A357638   .  .  .  (322)      (43)      (52)     (61)  (7)
%e A357638            (331)      (421)     (511)
%e A357638            (2221)     (3211)    (4111)
%e A357638            (1111111)  (22111)   (31111)
%e A357638                       (211111)
%t A357638 skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]),{i,Length[f]}];
%t A357638 Table[Length[Select[IntegerPartitions[n],skats[#]==k&]],{n,0,12},{k,-n,n,2}]
%Y A357638 Row sums are A000041.
%Y A357638 Number of nonzero entries in row n appears to be A004396(n+1).
%Y A357638 First nonzero entry of each row appears to converge to A146325.
%Y A357638 The central column is A035544, half A035363.
%Y A357638 Column sums appear to be A298311.
%Y A357638 For original alternating sum we have A344651, ordered A097805.
%Y A357638 The half-alternating version is A357637.
%Y A357638 The ordered version (compositions) is A357646, half A357645.
%Y A357638 The reverse version is A357705, half A357704.
%Y A357638 A351005 = alternately equal and unequal partitions, compositions A357643.
%Y A357638 A351006 = alternately unequal and equal partitions, compositions A357644.
%Y A357638 A357621 gives half-alternating sum of standard compositions, skew A357623.
%Y A357638 A357629 gives half-alternating sum of prime indices, skew A357630.
%Y A357638 A357633 gives half-alternating sum of Heinz partition, skew  A357634.
%Y A357638 Cf. A035594, A053251, A357136, A357189, A357486, A357487, A357488, A357624, A357631, A357632, A357636.
%K A357638 nonn,tabl
%O A357638 0,13
%A A357638 _Gus Wiseman_, Oct 10 2022