A322241 - OEIS

%I #4 Dec 08 2018 09:16:25

%S 1,1,41,249,6305,77569,1665321,27724889,574252417,10958980929,

%T 228679916905,4671350051321,99292476904609,2107949882690241,

%U 45658568907254505,993562984208479193,21876513296218002433,484448162130512673665,10812975015547281792937,242647271141110287979513,5477046865641884201456033

%N G.f.: exp( Sum_{n>=1} A084605(n)^2 * x^n/n ), where A084605(n) is the central coefficient in (1 + x + 4*x^2)^n.

%C Compare to: exp( Sum_{n>=1} A084605(n) * x^n/n ) = (1-x - sqrt(1 - 2*x - 15*x^2))/(8*x^2), the g.f. of A091147.

%C Sequence A322240(n) = A084605(n)^2 has generating function 1 / AGM(1 + 15*x, sqrt((1 - 9*x)*(1 - 25*x)) ).

%e G.f.: A(x) = 1 + x + 41*x^2 + 249*x^3 + 6305*x^4 + 77569*x^5 + 1665321*x^6 + 27724889*x^7 + 574252417*x^8 + 10958980929*x^9 + 228679916905*x^10 + ...

%e such that

%e log(A(x)) = x + 81*x^2/2 + 625*x^3/3 + 21025*x^4/4 + 314721*x^5/5 + 8071281*x^6/6 + 155975121*x^7/7 + 3685097025*x^8/8 + ... + A084605(n)^2 * x^n/n + ...

%e RELATED SERIES.

%e The g.f. of A084605 equals the series

%e 1/sqrt(1 - 2*x - 15*x^2) = 1 + x + 9*x^2 + 25*x^3 + 145*x^4 + 561*x^5 + 2841*x^6 + 12489*x^7 + 60705*x^8 + 281185*x^9 + ... + A084605(n) * x^n/n + ...

%o (PARI) {a(n)=if(n==0, 1, polcoeff(exp(sum(m=1, n, polcoeff(1/sqrt(1 - 2*x - 15*x^2 +x*O(x^m)), m)^2 *x^m/m)+x*O(x^n)), n))}

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

%Y Cf. A322240, A084605.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 08 2018