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%I #16 Nov 12 2022 09:42:25
%S 1,0,1,0,-1,1,0,1,-1,1,0,0,0,-1,1,0,-1,1,0,-1,1,0,0,0,0,0,-1,1,0,1,-1,
%T 1,0,0,-1,1,0,0,0,0,0,0,0,-1,1,0,0,0,-1,1,0,0,0,-1,1,0,0,0,0,0,0,0,0,
%U 0,-1,1,0,-1,1,0,-1,1,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,-1,1,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1
%N Inverse of number triangle A101688.
%C Row sums are the Fredholm-Rueppel sequence A036987 [conjecture].
%H R. J. Mathar, <a href="/A111967/a111967.pdf">OEIS A111967</a>
%F G.f. of k-th column is x^k*if(k=0,1,x*Sum_{j>=0} (-1)^j*x^(-2^(j/2)*(((k+2)/(2*sqrt(2))-(k+1))(-1)^j-(k+2)/(2*sqrt(2))-(k+1))-(k+2))+1-x). - _Paul Barry_, Jan 30 2007
%e Triangle begins
%e 1,
%e 0, 1,
%e 0, -1, 1,
%e 0, 1, -1, 1,
%e 0, 0, 0, -1, 1,
%e 0, -1, 1, 0, -1, 1,
%e 0, 0, 0, 0, 0, -1, 1,
%e 0, 1, -1, 1, 0, 0, -1, 1,
%e 0, 0, 0, 0, 0, 0, 0, -1, 1,
%e 0, 0, 0, -1, 1, 0, 0, 0, -1, 1,
%e 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1,
%e 0, -1, 1, 0, -1, 1, 0, 0, 0, 0, -1, 1,
%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1,
%e 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, -1, 1,
%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1,
%e 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, -1, 1,
%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1
%o (PARI) T(n, k) = if(binomial(k, n-k)>0, 1, 0); \\ A101688
%o mrepeat(nn) = matrix(nn, nn, n, k, T(n-1, k-1)); \\ A101688
%o lista(nn) = my(m=mrepeat(nn+1), im = 1/m, list = List()); for (n = 1, nn, listput(list, vector(n, k, im[n,k]));); Vec(list); \\ _Michel Marcus_, Nov 12 2022
%Y Cf. A036987, A101688
%K sign,tabl
%O 0,1
%A _Paul Barry_, Aug 23 2005
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