%I #9 Dec 07 2018 18:09:03

%S 1,3,15,76,357,1662,8203,36609,169800,788024,3586350,15948147,

%T 73761986,324147729,1454796651,6544916640,28902107643,126842754933,

%U 567156315794,2468434955040,10893525305088,47854663427104,208582052412240,905923236202737,3975385018556868,17200981327476354,74619131550054048,323976744392754994,1400917964875907424,6031485491299656747

%N G.f.: exp( Sum_{n>=1} A322185(n)*x^n/n ), where A322185(n) = sigma(2*n) * binomial(2*n,n)/2.

%C Related series:

%C (1) Product_{n>=1} (1 - x^(2*n))/(1 - x^n)^3 = exp( Sum_{n>=1} sigma(2*n) * x^n/n ) (see formula of Joerg Arndt in A182818).

%C (2) C(x) = exp( Sum_{n>=1} binomial(2*n,n)/2 * x^n/n ), where C(x) = 1 + x*C(x)^2 is the Catalan function (A000108).

%C A322185(n) is also the coefficient of x^n*y^n/n in log( Product_{n>=1} 1/(1 - (x + y)^n) ).

%H Paul D. Hanna, <a href="/A322186/b322186.txt">Table of n, a(n) for n = 0..512</a>

%e G.f.: A(x) = 1 + 3*x + 15*x^2 + 76*x^3 + 357*x^4 + 1662*x^5 + 8203*x^6 + 36609*x^7 + 169800*x^8 + 788024*x^9 + 3586350*x^10 + 15948147*x^11 + ...

%e such that

%e log(A(x)) = 3*x + 21*x^2/2 + 120*x^3/3 + 525*x^4/4 + 2268*x^5/5 + 12936*x^6/6 + 41184*x^7/7 + 199485*x^8/8 + 948090*x^9/9 + 3879876*x^10/10 + 12697776*x^11/11 + ... + A322185(n)*x^n/n + ...

%e RELATED SERIES.

%e A(x)^2 = 1 + 6*x + 39*x^2 + 242*x^3 + 1395*x^4 + 7746*x^5 + 42864*x^6 + 226560*x^7 + 1185417*x^8 + 6126642*x^9 + 31178598*x^10 + 156270312*x^11 + 780797727*x^12 + ...

%e where A(x)^2 = exp( Sum_{n>=1} sigma(2*n) * binomial(2*n,n) * x^n/n ).

%o (PARI) {A322185(n) = sigma(2*n) * binomial(2*n,n)/2}

%o {a(n) = polcoeff( exp( sum(m=1, n, A322185(m)*x^m/m ) +x*O(x^n) ), n) }

%o for(n=0, 30, print1( a(n), ", ") )

%Y Cf. A322185, A322204, A322188.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Dec 07 2018