A370143 - OEIS

%I #9 Feb 15 2024 08:21:40

%S 1,-4,16,-64,252,-980,3752,-14076,51384,-180488,597812,-1788936,

%T 4284828,-3665376,-47694524,475585724,-3186717720,18465627936,

%U -98993741736,504886869432,-2484393916472,11887606679816,-55566604265244,254411449305096,-1142492183274444,5033985939170544

%N Expansion of g.f. A(x) satisfying Sum_{n>=0} Product_{k=1..n} (x^k + 3*A(x)) = 1 + 4*Sum_{n>=1} x^(n*(n+1)/2).

%H Paul D. Hanna, <a href="/A370143/b370143.txt">Table of n, a(n) for n = 1..211</a>

%F G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

%F Let Q(x) = 1 + 4*Sum_{n>=1} x^(n*(n+1)/2), then

%F (1) Q(x) = Sum_{n>=0} Product_{k=1..n} (x^k + 3*A(x)).

%F (2) Q(x) = Sum_{n>=0} x^(n*(n+1)/2) / Product_{k=0..n} (1 - x^k * 3*A(x)).

%F (3) Q(x) = 1/(1 - F(1)), where F(n) = (x^n + 3*A(x))/(1 + x^n + 3*A(x) - F(n+1)), a continued fraction.

%e G.f.: A(x) = x - 4*x^2 + 16*x^3 - 64*x^4 + 252*x^5 - 980*x^6 + 3752*x^7 - 14076*x^8 + 51384*x^9 - 180488*x^10 + 597812*x^11 - 1788936*x^12 + ...

%e Let Q(x) = 1 + 4*Sum_{n>=1} x^(n*(n+1)/2)

%e then A = A(x) satisfies

%e (1) Q(x) = 1 + (x + 3*A) + (x + 3*A)*(x^2 + 3*A) + (x + 3*A)*(x^2 + 3*A)*(x^3 + 3*A) + (x + 3*A)*(x^2 + 3*A)*(x^3 + 3*A)*(x^4 + 3*A) + (x + 3*A)*(x^2 + 3*A)*(x^3 + 3*A)*(x^4 + 3*A)*(x^5 + 3*A) + ...

%e also

%e (2) Q(x) = 1/(1 - 3*A) + x/((1 - 3*A)*(1 - x*3*A)) + x^3/((1 - 3*A)*(1 - x*3*A)*(1 - x^2*3*A)) + x^6/((1 - 3*A)*(1 - x*3*A)*(1 - x^2*3*A)*(1 - x^3*3*A)) + x^10/((1 - 3*A)*(1 - x*3*A)*(1 - x^2*3*A)*(1 - x^3*3*A)*(1 - x^4*3*A)) + ...

%e Further, A = A(x) satisfies the continued fraction given by

%e (3) Q(x) = 1/(1 - (x + 3*A)/(1 + x + 3*A - (x^2 + 3*A)/(1 + x^2 + 3*A - (x^3 + 3*A)/(1 + x^3 + 3*A - (x^4 + 3*A)/(1 + x^4 + 3*A - (x^5 + 3*A)/(1 + x^5 + 3*A - (x^6 + 3*A)/(1 + x^6 + 3*A - (x^7 + 3*A)/(1 - ...)))))))).

%e where

%e Q(x) = 1 + 4*x + 4*x^3 + 4*x^6 + 4*x^10 + 4*x^15 + 4*x^21 + ... + 4*x^(n*(n+1)/2) + ...

%o (PARI) {a(n,y=3) = my(A=[0,1]); for(i=1,n, A = concat(A,0);

%o A[#A] = polcoeff( (sum(m=1,#A, prod(k=1,m, x^k + y*Ser(A) ) ) - (y+1)*sum(m=1,sqrtint(2*#A+1), x^(m*(m+1)/2) ) )/(-y), #A-1)  ); H=A; A[n+1]}

%o for(n=1,40, print1(a(n),", "))

%Y Cf. A370140, A370141, A370142, A370144.

%K sign

%O 1,2

%A _Paul D. Hanna_, Feb 14 2024