A370144 - OEIS

%I #10 Feb 15 2024 08:21:44

%S 1,-5,25,-125,620,-3055,14935,-72320,346120,-1632435,7555615,

%T -34103940,148616480,-614229500,2321723005,-7293541865,11847720800,

%U 80050256480,-1182000043580,10263664188460,-75379977023875,508865668922995,-3262422374486260,20187692749822600,-121673379897635840

%N Expansion of g.f. A(x) satisfying Sum_{n>=0} Product_{k=1..n} (x^k + 4*A(x)) = 1 + 5*Sum_{n>=1} x^(n*(n+1)/2).

%H Paul D. Hanna, <a href="/A370144/b370144.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 + 5*Sum_{n>=1} x^(n*(n+1)/2), then

%F (1) Q(x) = Sum_{n>=0} Product_{k=1..n} (x^k + 4*A(x)).

%F (2) Q(x) = Sum_{n>=0} x^(n*(n+1)/2) / Product_{k=0..n} (1 - x^k * 4*A(x)).

%F (3) Q(x) = 1/(1 - F(1)), where F(n) = (x^n + 4*A(x))/(1 + x^n + 4*A(x) - F(n+1)), a continued fraction.

%e G.f.: A(x) = x - 5*x^2 + 25*x^3 - 125*x^4 + 620*x^5 - 3055*x^6 + 14935*x^7 - 72320*x^8 + 346120*x^9 - 1632435*x^10 + 7555615*x^11 + ...

%e Let Q(x) = 1 + 5*Sum_{n>=1} x^(n*(n+1)/2)

%e then A = A(x) satisfies

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

%e also

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

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

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

%e where

%e Q(x) = 1 + 5*x + 5*x^3 + 5*x^6 + 5*x^10 + 5*x^15 + 5*x^21 + ... + 5*x^(n*(n+1)/2) + ...

%o (PARI) {a(n,y=4) = 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, A370143.

%K sign

%O 1,2

%A _Paul D. Hanna_, Feb 14 2024