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A126713 The triangle K referred to in A038200, read along rows.

%I A126713 #8 Dec 13 2013 09:23:10
%S A126713 1,-1,1,1,-3,1,-1,7,-4,1,1,-15,10,-5,1,-1,31,-19,15,-6,1,1,-63,28,-35,
%T A126713 21,-7,1,-1,127,-28,71,-56,28,-8,1,1,-255,1,-135,126,-84,36,-9,1,-1,
%U A126713 511,80,255,-251,210,-120,45,-10,1,1,-1023,-242,-495,451,-462,330,-165,55,-11,1,-1,2047,485,991,-726,925,-792,495,-220
%N A126713 The triangle K referred to in A038200, read along rows.
%C A126713 This means the description of A038200 is slightly incorrect and ought be: "Row sums of triangle K(m,n), inverse to a triangle obtained from A020921 after eliminating the leftmost column."
%F A126713 G.f.: (1/(1+x))*Sum(x^(k-1)/((1+x)^k-y*x^k),k=1..infinity). - _Vladeta Jovovic_, Feb 26 2008
%e A126713 If the leftmost column of the triangle in A020921 is deleted we get
%e A126713 1
%e A126713 1 1
%e A126713 2 3 1
%e A126713 2 5 4 1
%e A126713 4 10 10 5 1
%e A126713 2 11 19 15 6 1
%e A126713 6 21 35 35 21 7 1
%e A126713 4 22 52 69 56 28 8 1
%e A126713 6 33 83 126 126 84 36 9 1
%e A126713 The present triangle is the inverse of this, namely
%e A126713 1
%e A126713 -1 1
%e A126713 1 -3 1
%e A126713 -1 7 -4 1
%e A126713 1 -15 10 -5 1
%e A126713 -1 31 -19 15 -6 1
%e A126713 1 -63 28 -35 21 -7 1
%e A126713 -1 127 -28 71 -56 28 -8 1
%e A126713 with row sums 1,0,-1,3,-8,21,-54,134,-318,720 of A038200.
%p A126713 A020921 := proc(n,k) option remember; local divs; if n <= 0 then 1; elif k > n then 0; else divs := numtheory[divisors](n); add(numtheory[mobius](op(i,divs))*binomial(n/op(i,divs),k),i=1..nops(divs)); fi; end: A020921t := proc(n,k) option remember; A020921(n+1,k+1); end: TriLInv := proc(nmax) local a,row,col; a := array(0..nmax,0..nmax); for row from 0 to nmax do for col from row+1 to nmax do a[row,col] := 0; od; od; for row from 0 to nmax do for col from row to 0 by -1 do if row <> col then a[row,col] := -add(a[row,c]*A020921t(c,col),c=col+1..row)/A020921t(col,col); else a[row,col] := (1-add(a[row,c]*A020921t(c,col),c=col+1..row))/A020921t(col,col); fi; od; od; RETURN(a); end: nmax := 12 : a := TriLInv(nmax) : for row from 0 to nmax do for col from 0 to row do printf("%d, ",a[row,col]); od; od:
%t A126713 f[n_] := (1/(1+x))*Sum[x^(k-1)/((1+x)^k-y*x^k), {k, 1, n+1}]; t[0, 0] = 1; t[n_, k_] := SeriesCoefficient[f[n], {x, 0, n}, {y, 0, k}]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Dec 13 2013, after _Vladeta Jovovic_ *)
%Y A126713 Cf. A039912.
%K A126713 sign,tabl
%O A126713 0,5
%A A126713 _R. J. Mathar_, Feb 12 2007