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A124448 Riordan array (sqrt(1+4x^2)-2x, (1+2x-sqrt(1+4x^2))/2).

%I A124448 #21 Jan 26 2020 21:49:15
%S A124448 1,-2,1,2,-3,1,0,4,-4,1,-2,-1,7,-5,1,0,-4,-4,11,-6,1,4,2,-6,-10,16,-7,
%T A124448 1,0,8,8,-6,-20,22,-8,1,-10,-5,11,19,-1,-35,29,-9,1,0,-20,-20,7,34,13,
%U A124448 -56,37,-10,1,28,14,-26,-46,-12,49,41,-84
%N A124448 Riordan array (sqrt(1+4x^2)-2x, (1+2x-sqrt(1+4x^2))/2).
%C A124448 Inverse of triangle A106195.
%C A124448 Row sums are A105523 (expansion of 1-xc(-x^2) where c(x) is the g.f. of A000108).
%C A124448 Product of A007318 and A124448 is inverse of A053538.
%C A124448 A124448*A007318 = A106180, as infinite lower triangular matrices. - _Philippe Deléham_, Oct 16 2007
%C A124448 Triangle T(n,k), read by rows, given by (-2,1,-1,1,-1,1,-1,1,-1,...) DELTA (1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Oct 09 2011
%e A124448 Triangle begins
%e A124448    1;
%e A124448   -2,   1;
%e A124448    2,  -3,   1;
%e A124448    0,   4,  -4,   1;
%e A124448   -2,  -1,   7,  -5,   1;
%e A124448    0,  -4,  -4,  11,  -6,   1;
%e A124448    4,   2,  -6, -10,  16,  -7,   1;
%e A124448    0,   8,   8,  -6, -20,  22,  -8,   1;
%o A124448 (PARI)
%o A124448 N=12;
%o A124448 T(n, k)=sum(i=0, n-k, binomial(k, i)*binomial(n-k, i)*2^(n-k-i));
%o A124448 M=matrix(N, N);
%o A124448 for(n=1, N, for(k=1, n, M[n, k]=T(n-1, k-1))); /* A106195 */
%o A124448 A=M^-1;  /* A124448 */
%o A124448 /* for (n=1, N, for(k=1, n, print1(M[n, k], ", "))); */ /* A106195 */
%o A124448 for (n=1, N, for(k=1, n, print1(A[n, k], ", "))); /* A124448 */
%o A124448 /* _Joerg Arndt_, May 14 2011 */
%Y A124448 Cf. A106195, A000045, A164948, A164942.
%K A124448 easy,sign,tabl
%O A124448 0,2
%A A124448 _Paul Barry_, Nov 01 2006
%E A124448 Edited by _N. J. A. Sloane_, Dec 29 2011