A197601 - OEIS

%I #32 May 29 2022 07:24:10

%S 1,1,5,14,52,187,708,2734,10758,43004,174004,711660,2936564,12211688,

%T 51124185,215299685,911445413,3876523626,16556573129,70980163570,

%U 305343924258,1317634326631,5702146948069,24741071869651,107608326588838,469073933764287

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

%H Vaclav Kotesovec, <a href="/A197601/b197601.txt">Table of n, a(n) for n = 0..1000</a>

%F 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 Logarithmic derivative equals A198059.

%e G.f.: A(x) = 1 + x + 5*x^2 + 14*x^3 + 52*x^4 + 187*x^5 + 708*x^6 +...

%e The logarithm of the g.f. begins:

%e 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 and equals the sum of the series:

%e log(A(x)) = (1 + 2^2*x + x^2)*x

%e + (1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4)*x^2/2

%e + (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 + (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 + (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 which involves the squares of the coefficients in even powers of (1+x).

%e The logarithm of the g.f. can also be expressed as:

%e log(A(x)) = (1-x)^5*(1 + 3^2*x + 6^2*x^2 + 10^2*x^3 + 15^2*x^4 +...)*x

%e + (1-x)^9*(1 + 5^2*x + 15^2*x^2 + 35^2*x^3 + 70^2*x^4 +...)*x^2/2

%e + (1-x)^13*(1 + 7^2*x + 28^2*x^2 + 84^2*x^3 + 210^2*x^4 +...)*x^3/3

%e + (1-x)^17*(1 + 9^2*x + 45^2*x^2 + 165^2*x^3 + 495^2*x^4 +...)*x^4/4

%e + (1-x)^21*(1 + 11^2*x + 66^2*x^2 + 286^2*x^3 + 1001^2*x^4 +...)*x^5/5 +...

%e which involves the squares of the coefficients in odd powers of 1/(1-x).

%t 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 (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 (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 Cf. A198059 (log), A186236, A004148, A180717, A180718.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Oct 20 2011