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Logarithmic derivative of A180718.
1

%I #2 Mar 30 2012 18:37:22

%S 1,5,16,61,226,884,3543,14429,59623,248950,1049159,4454356,19032976,

%T 81769735,352967821,1529948477,6655903632,29050257899,127162016206,

%U 558088733406,2455157735151,10824115727199,47814658900427

%N Logarithmic derivative of A180718.

%F L.g.f.: L(x) = Sum_{n>=0} [ Sum_{k=0..n} C(n,k)^2*x^k ]^2*x^n/n.

%e L.g.f.: L(x) = x + 5*x^2/2 + 16*x^3/3 + 61*x^4/4 + 226*x^5/5 +...

%e which equals the sum of the series:

%e L(x) = (1 + x)^2*x

%e + (1 + 4*x + x^2)^2*x^2/2

%e + (1 + 9*x + 9*x^2 + x^3)^2*x^3/3

%e + (1 + 16*x + 36*x^2 + 16*x^3 + x^4)^2*x^4/4

%e + (1 + 25*x + 100*x^2 + 100*x^3 + 25*x^4 + x^5)^2*x^5/5

%e + (1 + 36*x + 225*x^2 + 400*x^3 + 225*x^4 + 36*x^5 + x^6)^2*x^6/6 +...

%e where exponentiation yields the integer series:

%e exp(L(x)) = 1 + x + 3*x^2 + 8*x^3 + 25*x^4 + 80*x^5 + 271*x^6 + 952*x^7 + 3443*x^8 + 12758*x^9 + 48212*x^10 +...+ A180718(n)*x^n/n +...

%o (PARI) {a(n)=n*polcoeff(sum(m=1,n,sum(k=0,m,binomial(m,k)^2*x^k)^2*x^m/m)+x*O(x^n),n)}

%Y Cf. A180718.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Sep 24 2010