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A125276 Eigensequence of triangle A039598: a(n) = Sum_{k=0..n-1} A039598(n-1,k)*a(k) for n>0 with a(0)=1.

%I A125276 #7 Dec 09 2013 06:24:02
%S A125276 1,1,3,12,58,325,2060,14514,112170,941128,8502393,82160481,844532873,
%T A125276 9191329357,105491177081,1272418794619,16080824798705,212370154398094,
%U A125276 2923859710010527,41877072960374478,622763691600244335
%N A125276 Eigensequence of triangle A039598: a(n) = Sum_{k=0..n-1} A039598(n-1,k)*a(k) for n>0 with a(0)=1.
%H A125276 Vaclav Kotesovec, <a href="/A125276/b125276.txt">Table of n, a(n) for n = 0..500</a>
%F A125276 a(n) = Sum_{k=0..n-1} a(k) * C(2*n,n-k-1)*(k+1)/n for n>0 with a(0)=1.
%F A125276 G.f. A(x) satisfies: A(x/(1+x)^2) = 1 + x*A(x); also, A(x*(1-x)) = 1 + [x/(1-x)]*A(x/(1-x)); also, A(x) = 1 + x*C(x)^2*A(x*C(x)^2) where C(x) = (1 - sqrt(1-4x))/(2x) is the Catalan function (A000108). - _Paul D. Hanna_, Aug 15 2007
%e A125276 a(3) = 5*(1) + 4*(1) + 1*(3) = 12;
%e A125276 a(4) = 14*(1) + 14*(1) + 6*(3) + 1*(12) = 58;
%e A125276 a(5) = 42*(1) + 48*(1) + 27*(3) + 8*(12) + 1*(58) = 325.
%e A125276 Triangle A039598(n,k) = C(2*n+2,n-k)*(k+1)/(n+1) begins:
%e A125276 1;
%e A125276 2, 1;
%e A125276 5, 4, 1;
%e A125276 14, 14, 6, 1;
%e A125276 42, 48, 27, 8, 1;
%e A125276 132, 165, 110, 44, 10, 1; ...
%e A125276 where g.f. of column k = G000108(x)^(2*k+2)
%e A125276 and G000108(x) = (1 - sqrt(1-4*x))/(2x) is the Catalan function.
%t A125276 A125276=ConstantArray[0,20]; A125276[[1]]=1; Do[A125276[[n]]=Binomial[2*n,n-1]/n+Sum[A125276[[k]]*Binomial[2*n,n-k-1]*(k+1)/n,{k,1,n-1}];,{n,2,20}]; Flatten[{1,A125276}] (* _Vaclav Kotesovec_, Dec 09 2013 *)
%o A125276 (PARI) a(n)=if(n==0,1,sum(k=0,n-1, a(k)*binomial(2*n, n-k-1)*(k+1)/n))
%Y A125276 Cf. A039598, A000108; A125275 (variant).
%K A125276 nonn
%O A125276 0,3
%A A125276 _Paul D. Hanna_, Nov 26 2006