login
G.f. A(x) satisfies: 1/(1-x) = Sum_{n>=0} x^n * (1+x)^(n^2) / A(x)^(n*(n+1)/2).
2

%I #9 Jun 01 2019 06:20:40

%S 1,1,1,1,1,2,4,13,41,152,585,2408,10398,46922,220594,1075614,5429158,

%T 28296558,152028318,840671002,4778156476,27882129540,166867893396,

%U 1023291150769,6424498576966,41262660744958,270921037479716,1817215371471834,12444490783066300,86956014104628146,619630222992962304,4500360360596423580,33298628793663193094,250878144285496830342,1923797779018220870457,15008171270194837029808

%N G.f. A(x) satisfies: 1/(1-x) = Sum_{n>=0} x^n * (1+x)^(n^2) / A(x)^(n*(n+1)/2).

%C Compare to: 1+x = Sum_{n>=0} x^n * (1+x)^(n^2) / G(x)^(n*(n+1)/2) holds when G(x) = (1+x)^2.

%H Paul D. Hanna, <a href="/A325578/b325578.txt">Table of n, a(n) for n = 0..300</a>

%e G.f.: A(x) = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + 4*x^6 + 13*x^7 + 41*x^8 + 152*x^9 + 585*x^10 + 2408*x^11 + 10398*x^12 + 46922*x^13 + 220594*x^14 + ...

%e such that

%e 1/(1-x) = 1 + x*(1+x)/A(x) + x^2*(1+x)^4/A(x)^3 + x^3*(1+x)^9/A(x)^6 + x^4*(1+x)^16/A(x)^10 + x^5*(1+x)^25/A(x)^15 + x^6*(1+x)^36/A(x)^21 + x^7*(1+x)^49/A(x)^28 + x^8*(1+x)^64/A(x)^36 + x^9*(1+x)^81/A(x)^45 + ...

%o (PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);

%o A[#A] = polcoeff( sum(m=0, #A, x^m*((1+x +x*O(x^#A))^(m^2)/Ser(A)^(m*(m+1)/2) - 1)), #A) ); A[n+1]}

%o for(n=0, 40, print1(a(n), ", "))

%Y Cf. A325579.

%K nonn

%O 0,6

%A _Paul D. Hanna_, May 22 2019