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A114476 Triangle read by rows: inverse of triangle in A061554 with signs in each column +,+,-,-,+,+,-,-,...

%I A114476 #14 May 01 2023 13:10:23
%S A114476 1,-1,1,3,-1,1,-3,4,-1,1,5,-4,5,-1,1,-5,9,-5,6,-1,1,7,-9,14,-6,7,-1,1,
%T A114476 -7,16,-14,20,-7,8,-1,1,9,-16,30,-20,27,-8,9,-1,1,-9,25,-30,50,-27,35,
%U A114476 -9,10,-1,1,11,-25,55,-50,77,-35,44,-10,11,-1,1,-11,36,-55,105,-77,112,-44,54,-11,12,-1,1,13,-36,91,-105,182,-112,156,-54,65,-12,13,-1,1
%N A114476 Triangle read by rows: inverse of triangle in A061554 with signs in each column +,+,-,-,+,+,-,-,...
%C A114476 Unsigned row sums appear to be A014739.
%e A114476 Start with a signed version of A061554:
%e A114476 1;
%e A114476 1, 1;
%e A114476 -2, 1, 1;
%e A114476 -3, -3, 1, 1;
%e A114476 6, -4, -4, 1, 1;
%e A114476 10, 10, -5, -5, 1;
%e A114476 ...
%e A114476 and invert it, getting:
%e A114476 1
%e A114476 -1, 1;
%e A114476 3, -1, 1;
%e A114476 -3, 4, -1, 1;
%e A114476 5, -4, 5, -1, 1;
%e A114476 -5, 9, -5, 6, -1, 1;
%e A114476 ...
%p A114476 A061554 := proc(n,k) binomial(n+k,floor(k/2)) ; end: nmax := 13 : A := array(1..nmax,1..nmax) : for r from 1 to nmax do for c from 1 to nmax do A[r,c] := A061554(c-1,r-c)*(-1)^floor((r-c)/2) ; od: od: A := linalg[inverse](A) : for r from 1 to nmax do for c from 1 to r do printf("%d, ", A[r,c]) ; od: od: # _R. J. Mathar_, Jan 31 2008
%t A114476 A061554[n_, k_] := Binomial[n+k, Floor[k/2]];
%t A114476 nmax = 13;
%t A114476 A = Table[A061554[c-1, r-c]*(-1)^Floor[(r-c)/2], {r, nmax}, {c, nmax}];
%t A114476 A = Inverse[A];
%t A114476 Table[A[[r, c]], {r, nmax}, {c, r}] // Flatten (* _Jean-François Alcover_, May 01 2023 *)
%K A114476 sign,tabl,easy
%O A114476 0,4
%A A114476 _Gary W. Adamson_, Nov 27 2006
%E A114476 Edited by _N. J. A. Sloane_, Dec 01 2006
%E A114476 More terms from _R. J. Mathar_, Jan 31 2008