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A187038 Row sums of number triangle A187037.

%I A187038 #18 Sep 30 2018 10:41:54
%S A187038 1,2,3,2,0,2,5,2,0,2,2,2,0,2,7,2,0,2,2,2,0,2,4,2,0,2,2,2,0,2,9,2,0,2,
%T A187038 2,2,0,2,4,2,0,2,2,2,0,2,6,2,0,2,2,2,0,2,4,2,0,2,2,2,0,2,11,2,0,2,2,2,
%U A187038 0,2,4,2,0,2,2,2,0,2,6,2,0,2,2,2,0,2,4,2,0,2,2,2,0,2,8,2,0,2,2,2,0,2,4,2,0,2
%N A187038 Row sums of number triangle A187037.
%C A187038 Apparently, apart from signs, same as A269735 (with a shift). If so, the g.f. for this sequence is obtained from that for A269735 by replacing x by -x. - _N. J. A. Sloane_, Mar 11 2016
%C A187038 For construction, see Barry, 2011. Although the paper doesn't treat especially this sequence, it outlines a general method for creating such sequences. - _Antti Karttunen_, Sep 30 2018
%H A187038 Antti Karttunen, <a href="/A187038/b187038.txt">Table of n, a(n) for n = 0..1025</a>
%H A187038 P. Barry, <a href="http://arxiv.org/abs/1107.5490">Invariant number triangles, eigentriangles and Somos-4 sequences</a>, arXiv:1107.5490 [math.CO], 2011.
%o A187038 (PARI)
%o A187038 up_to = 128;
%o A187038 A187034aux(n,k) = if(k>n,0,if(n<=2*k, (-1)^(n-k), 0));
%o A187038 A187034downshifted_and_negated(n,k) = if(k==n,1,-A187034aux(n-1,k));
%o A187038 A187038list(up_to) = { my(m1=matrix(up_to,up_to,n,k,A187034downshifted_and_negated(n-1,k-1)), m2 = matsolve(m1,matid(up_to)), v = vector(up_to)); for(n=1,up_to,v[n] = vecsum(m2[n,])); (v); };
%o A187038 write_A187036_and_A187038list(up_to) = { my(m1=matrix(up_to,up_to,n,k,A187034downshifted_and_negated(n-1,k-1)), m2 = matsolve(m1,matid(up_to)), v187036 = (m2[,1]~), v187038 = vector(up_to,j,vecsum(m2[j,]))); for(n=1,up_to,write("b187036.txt", n-1, " ", v187036[n]); write("b187038.txt", n-1, " ", v187038[n])); }; \\ For computing both at the same time
%o A187038 v187038 = A187038list(1+up_to);
%o A187038 A187038(n) = v187038[1+n]; \\ _Antti Karttunen_, Sep 29 2018
%Y A187038 Cf. A187034, A187036, A187037, A269735.
%K A187038 nonn
%O A187038 0,2
%A A187038 _Paul Barry_, Mar 08 2011
%E A187038 More terms from _Antti Karttunen_, Sep 29 2018