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A113187 Inverse of twin-prime related triangle A111125.

%I A113187 #19 Feb 16 2017 03:19:20
%S A113187 1,-3,1,10,-5,1,-35,21,-7,1,126,-84,36,-9,1,-462,330,-165,55,-11,1,
%T A113187 1716,-1287,715,-286,78,-13,1,-6435,5005,-3003,1365,-455,105,-15,1,
%U A113187 24310,-19448,12376,-6188,2380,-680,136,-17,1,-92378,75582,-50388,27132,-11628,3876,-969,171,-19,1,352716,-293930,203490
%N A113187 Inverse of twin-prime related triangle A111125.
%C A113187 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 A113187 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 A113187 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 A113187 Indranil Ghosh, <a href="/A113187/b113187.txt">Rows 0..125 of triangle, flattened</a>
%F A113187 Riordan array ((sqrt(1+4x)-1)/(2x*sqrt(1+4x)), (1+2x-sqrt(1+4x))/(2x)).
%F A113187 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 A113187 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 A113187 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 A113187 Triangle T(n,k) begins:
%e A113187 n\k     0      1      2     3      4    5    6   7   8  9 ...
%e A113187 0:      1
%e A113187 1:     -3      1
%e A113187 2:     10     -5      1
%e A113187 3:    -35     21     -7     1
%e A113187 4:    126    -84     36    -9      1
%e A113187 5:   -462    330   -165    55    -11    1
%e A113187 6:   1716  -1287    715  -286     78  -13    1
%e A113187 7:  -6435   5005  -3003  1365   -455  105  -15   1
%e A113187 8:  24310 -19448  12376 -6188   2380 -680  136 -17   1
%e A113187 9: -92378  75582 -50388 27132 -11628 3876 -969 171 -19  1
%e A113187 ... Reformatted by Wolfdieter Lang, Oct 17 2012
%e A113187 From Wolfdieter Lang, Oct 18 2012: (Start)
%e A113187 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 A113187 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 A113187 easy,sign,tabl
%O A113187 0,2
%A A113187 _Paul Barry_, Oct 17 2005