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%I #8 Jul 19 2016 11:04:46
%S 1,2,12,148,3105,99156,4481449,272312216,21414443481,2116193061340,
%T 256712977920256,37506637787774112,6496315164318118165,
%U 1316230822119433518312,308426950979497974254310
%N Central terms in even-indexed rows of triangle A105556 and thus equals the n-th row sum of the n-th matrix power of the Narayana triangle A001263.
%C Each term a(n) is divisible by (n+1) for all n>=0.
%F Contribution from _Paul D. Hanna_, Jan 31 2009: (Start)
%F a(n) = (n+1)*A155926(n) for n>=0.
%F G.f.: A(x) = d/dx x*F(x) where F(x) = B(x*F(x)) and F(x) = Sum_{n>=0} A155926(n)*x^n/[n!*(n+1)!/2^n] with B(x) = Sum_{n>=0} x^n/[n!*(n+1)!/2^n] and A(x) = Sum_{n>=0} a(n)*x^n/[n!*(n+1)!/2^n]. (End)
%e Terms a(n) divided by (n+1) begin:
%e 1,1,4,37,621,16526,640207,34039027,2379382609,211619306134,...
%e Contribution from _Paul D. Hanna_, Jan 31 2009: (Start)
%e G.f.: A(x) = 1 + 2*x + 12*x^2/3 + 148*x^3/18 + 3105*x^4/180 +...+ a(n)*x^n/[n!*(n+1)!/2^n] +...
%e G.f.: A(x) = d/dx x*F(x) where F(x) = B(x*F(x)) and:
%e F(x) = 1 + x + 4*x^2/3 + 37*x^3/18 + 621*x^4/180 + 16526*x^5/2700 +...+ A155926(n)*x^n/[n!*(n+1)!/2^n] +...
%e B(x) = 1 + x + x^2/3 + x^3/18 + x^4/180 +...+ x^n/[n!*(n+1)!/2^n] +... (End)
%o (PARI) a(n)=local(N=matrix(n+1,n+1,m,j,if(m>=j, binomial(m-1,j-1)*binomial(m,j-1)/j))); sum(j=0,n,(N^n)[n+1,j+1])
%o for(n=0,20,print1(a(n),", "))
%o (PARI) a(n)=local(F=sum(k=0,n,x^k/(k!*(k+1)!/2^k))+x*O(x^n));polcoeff(deriv(serreverse(x/F)),n)*n!*(n+1)!/2^n
%o for(n=0,20,print1(a(n),", ")) \\ _Paul D. Hanna_, Jan 31 2009
%Y Cf. A001263, A105556, A155926.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Apr 14 2005
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