cp's OEIS Frontend

A357642 Number of even-length integer compositions of 2n whose half-alternating sum is 0.

%I A357642 #13 Oct 13 2022 06:36:31
%S A357642 1,0,1,4,13,48,186,712,2717,10432,40222,155384,601426,2332640,9063380,
%T A357642 35269392,137438685,536257280,2094786870,8191506136,32063203590,
%U A357642 125613386912,492516592620,1932569186288,7588478653938,29816630378368,117226929901676,461151757861552
%N A357642 Number of even-length integer compositions of 2n whose half-alternating sum is 0.
%C A357642 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 A357642 David A. Corneth, <a href="/A357642/b357642.txt">Table of n, a(n) for n = 0..1665</a>
%e A357642 The a(0) = 1 through a(4) = 13 compositions:
%e A357642   ()  .  (1111)  (1212)  (1313)
%e A357642                  (1221)  (1322)
%e A357642                  (2112)  (1331)
%e A357642                  (2121)  (2213)
%e A357642                          (2222)
%e A357642                          (2231)
%e A357642                          (3113)
%e A357642                          (3122)
%e A357642                          (3131)
%e A357642                          (111311)
%e A357642                          (112211)
%e A357642                          (113111)
%e A357642                          (11111111)
%t A357642 Table[Length[Select[Join @@ Permutations/@IntegerPartitions[2n],EvenQ[Length[#]]&&halfats[#]==0&]],{n,0,9}]
%o A357642 (PARI) a(n) = {my(v, res); if(n < 3, return(1 - bitand(n,1))); res = 0; v = vector(2*n, i, binomial(n-1,i-1)); forstep(i = 4, 2*n, 2, lp = i\4 * 2; rp = i - lp; res += v[lp] * v[rp]; ); res } \\ _David A. Corneth_, Oct 13 2022
%Y A357642 The skew-alternating version appears to be A000984.
%Y A357642 For original alternating sum we have A001700/A088218.
%Y A357642 The version for partitions of any length is A357639, ranked by A357631.
%Y A357642 For length multiple of 4 we have A110145.
%Y A357642 These compositions of any length are ranked by A357625, reverse A357626.
%Y A357642 A124754 gives alternating sum of standard compositions, reverse A344618.
%Y A357642 A357621 = half-alternating sum of standard compositions, reverse A357622.
%Y A357642 A357637 counts partitions by half-alternating sum, skew A357638.
%Y A357642 Cf. A000583, A001511, A035363, A053251, A344619, A357136, A357182, A357627, A357628, A357629, A357633.
%K A357642 nonn
%O A357642 0,4
%A A357642 _Gus Wiseman_, Oct 12 2022
%E A357642 More terms from _Alois P. Heinz_, Oct 12 2022