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%I #19 Feb 16 2017 03:19:20
%S 1,-3,1,10,-5,1,-35,21,-7,1,126,-84,36,-9,1,-462,330,-165,55,-11,1,
%T 1716,-1287,715,-286,78,-13,1,-6435,5005,-3003,1365,-455,105,-15,1,
%U 24310,-19448,12376,-6188,2380,-680,136,-17,1,-92378,75582,-50388,27132,-11628,3876,-969,171,-19,1,352716,-293930,203490
%N Inverse of twin-prime related triangle A111125.
%C Row sums are (-1)^n*A000984. Diagonal sums are (-1)^n*A014301(n+1). An interesting factorization is (1/sqrt(1+4x)),(sqrt(1+4x)-1)/2)(1/(1+x),x/(1+x)).
%C The Z-sequence for this Riordan array is [-3,1], and the A-sequence is [1,-2,1]. For the Z- and A-sequence of Riordan arrays see the W. Lang link, with references, under A006232. - _Wolfdieter Lang_, Oct 18 2012
%C This triangle appears in the formula (x-1/x)^(2*n+1) = sum(T(n,k)*(x^(2*k+1) - 1/x^(2*k+1)),k=0..n), n >= 0. Proof from the inversion of the formula given in an Oct 18 2012 comment on A111125, due to the Riordan property. - _Wolfdieter Lang_, Nov 14 2012
%H Indranil Ghosh, <a href="/A113187/b113187.txt">Rows 0..125 of triangle, flattened</a>
%F Riordan array ((sqrt(1+4x)-1)/(2x*sqrt(1+4x)), (1+2x-sqrt(1+4x))/(2x)).
%F T(n, k)=(-1)^(n-k)*C(2n+1, n+k+1); T(n, k)=sum{j=0..n, (-1)^(n-k)*C(2n-j, n-j)C(j, k)}.
%F O.g.f. column k: ((2-c(-x))/(1+4*x))*(1-c(-x))^k, with the o.g.f. c(x) of A000108 (Catalan), k>=0. From the Riordan property given above. - _Wolfdieter Lang_, Oct 17 2012
%F O.g.f. of the row polynomials R(n,x) = sum(T(n,k)*x^k,k=0..n): ((2-c(-z))/(1+4*z))/(1-x*(1-c(-z))) = 1/((1+4*z)*(x-(1-x)^2*z))*(x+2*x*z-2*z + (1+x)*z*c(-z)), with the o.g.f. c(x) of A000108. - _Wolfdieter Lang_, Oct 18 2012
%e Triangle T(n,k) begins:
%e n\k 0 1 2 3 4 5 6 7 8 9 ...
%e 0: 1
%e 1: -3 1
%e 2: 10 -5 1
%e 3: -35 21 -7 1
%e 4: 126 -84 36 -9 1
%e 5: -462 330 -165 55 -11 1
%e 6: 1716 -1287 715 -286 78 -13 1
%e 7: -6435 5005 -3003 1365 -455 105 -15 1
%e 8: 24310 -19448 12376 -6188 2380 -680 136 -17 1
%e 9: -92378 75582 -50388 27132 -11628 3876 -969 171 -19 1
%e ... Reformatted by Wolfdieter Lang, Oct 17 2012
%e From Wolfdieter Lang, Oct 18 2012: (Start)
%e Recurrence from the Z-sequence [-3,1] (see a comment above): T(3,0) = -3*T(2,0) + 1*T(2,1) = -3*10 + (-5) = -35.
%e Recurrence from the A-sequence [1,-2,1]: T(5,1) = 1*T(4,0) -2*T(4,1) + 1*T(4,2) = 126 -2*(-84) +36 = 330. (End)
%K easy,sign,tabl
%O 0,2
%A _Paul Barry_, Oct 17 2005
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