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A197601 G.f.: exp( Sum_{n>=1} [Sum_{k=0..2*n} C(2*n,k)^2 *x^k] *x^n/n ).

%I A197601 #32 May 29 2022 07:24:10
%S A197601 1,1,5,14,52,187,708,2734,10758,43004,174004,711660,2936564,12211688,
%T A197601 51124185,215299685,911445413,3876523626,16556573129,70980163570,
%U A197601 305343924258,1317634326631,5702146948069,24741071869651,107608326588838,469073933764287
%N A197601 G.f.: exp( Sum_{n>=1} [Sum_{k=0..2*n} C(2*n,k)^2 *x^k] *x^n/n ).
%H A197601 Vaclav Kotesovec, <a href="/A197601/b197601.txt">Table of n, a(n) for n = 0..1000</a>
%F A197601 G.f.: exp( Sum_{n>=1} (1-x)^(4*n+1) *[Sum_{k>=0} C(2*n+k,k)^2 *x^k] *x^n/n ).
%F A197601 Logarithmic derivative equals A198059.
%e A197601 G.f.: A(x) = 1 + x + 5*x^2 + 14*x^3 + 52*x^4 + 187*x^5 + 708*x^6 +...
%e A197601 The logarithm of the g.f. begins:
%e A197601 log(A(x)) = x + 9*x^2/2 + 28*x^3/3 + 121*x^4/4 + 496*x^5/5 + 2100*x^6/6 + 9017*x^7/7 + 38969*x^8/8 +...+ A198059(n)*x^n/n +...
%e A197601 and equals the sum of the series:
%e A197601 log(A(x)) = (1 + 2^2*x + x^2)*x
%e A197601 + (1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4)*x^2/2
%e A197601 + (1 + 6^2*x + 15^2*x^2 + 20^2*x^3 + 15^2*x^4 + 6^2*x^5 + x^6)*x^3/3
%e A197601 + (1 + 8^2*x + 28^2*x^2 + 56^2*x^3 + 70^2*x^4 + 56^2*x^5 + 28^2*x^6 + 8^2*x^7 + x^8)*x^4/4
%e A197601 + (1 + 10^2*x + 45^2*x^2 + 120^2*x^3 + 210^2*x^4 + 252^2*x^5 + 210^2*x^6 + 120^2*x^7 + 45^2*x^8 + 10^2*x^9 + x^10)*x^5/5 +...
%e A197601 which involves the squares of the coefficients in even powers of (1+x).
%e A197601 The logarithm of the g.f. can also be expressed as:
%e A197601 log(A(x)) = (1-x)^5*(1 + 3^2*x + 6^2*x^2 + 10^2*x^3 + 15^2*x^4 +...)*x
%e A197601 + (1-x)^9*(1 + 5^2*x + 15^2*x^2 + 35^2*x^3 + 70^2*x^4 +...)*x^2/2
%e A197601 + (1-x)^13*(1 + 7^2*x + 28^2*x^2 + 84^2*x^3 + 210^2*x^4 +...)*x^3/3
%e A197601 + (1-x)^17*(1 + 9^2*x + 45^2*x^2 + 165^2*x^3 + 495^2*x^4 +...)*x^4/4
%e A197601 + (1-x)^21*(1 + 11^2*x + 66^2*x^2 + 286^2*x^3 + 1001^2*x^4 +...)*x^5/5 +...
%e A197601 which involves the squares of the coefficients in odd powers of 1/(1-x).
%t A197601 nmax = 30; CoefficientList[Series[Exp[Sum[Hypergeometric2F1[-2*k, -2*k, 1, x]*x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, May 29 2022 *)
%o A197601 (PARI) {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, n, binomial(2*m,k)^2 *x^k) *x^m/m)+x*O(x^n)), n)}
%o A197601 (PARI) {a(n)=polcoeff(exp(sum(m=1, n, (1-x+x*O(x^n))^(4*m+1) *sum(k=0, n-m+1, binomial(2*m+k, k)^2 *x^k+x*O(x^n)) *x^m/m)+x*O(x^n)), n)}
%Y A197601 Cf. A198059 (log), A186236, A004148, A180717, A180718.
%K A197601 nonn
%O A197601 0,3
%A A197601 _Paul D. Hanna_, Oct 20 2011