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A114188 Riordan array (1/(1-x),x(1+x)/(1-x)^2).

%I A114188 #15 Apr 21 2015 04:47:20
%S A114188 1,1,1,1,4,1,1,9,7,1,1,16,26,10,1,1,25,70,52,13,1,1,36,155,190,87,16,
%T A114188 1,1,49,301,553,403,131,19,1,1,64,532,1372,1462,736,184,22,1,1,81,876,
%U A114188 3024,4446,3206,1216,246,25,1,1,100,1365,6084,11826,11584,6190,1870,317
%N A114188 Riordan array (1/(1-x),x(1+x)/(1-x)^2).
%C A114188 Product of A007318 and A113953, that is, (1/(1-x),x/(1-x))*(1,x(1+2x)).
%C A114188 Row sums are A025192. Diagonal sums are A052980.
%C A114188 Inverse is A114189. A signed version is A110511.
%H A114188 P. Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Barry2/barry231.html">A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays</a>, Journal of Integer Sequences, 16 (2013), #13.5.4.
%F A114188 T(n, k) = Sum_{j=0..n} C(n, j)*C(k, j-k)2^(j-k).
%F A114188 T(n, k) = Sum_{j=0..n-k} C(k, j)*C(n+k-j, 2k).
%F A114188 T(n,k) = 2*T(n-1,k)+T(n-1,k-1)-T(n-2,k)+T(n-2,k-1), T(0,0)=T(1,0)=T(1,1)=1, T(n,k)=0 if k<0 or if k>n. - _Philippe Deléham_, Jan 11 2014
%F A114188 G.f.: 1/(1-y-x*(1+y)/(1-y)). - _Vladimir Kruchinin_, Apr 21 2015
%e A114188 Triangle begins
%e A114188 1;
%e A114188 1, 1;
%e A114188 1, 4, 1;
%e A114188 1, 9, 7, 1;
%e A114188 1, 16, 26, 10, 1;
%e A114188 1, 25, 70, 52, 13, 1;
%e A114188 1, 36,155,190, 87, 16, 1;
%Y A114188 Cf. A007318, A025192, A052980, A110511, A113953, A114189.
%K A114188 easy,nonn,tabl
%O A114188 0,5
%A A114188 _Paul Barry_, Nov 16 2005