cp's OEIS Frontend

A355354 G.f. A(x) satisfies: 4*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

%I A355354 #5 Jun 30 2022 10:40:07
%S A355354 1,4,28,196,1416,10860,87392,727188,6196212,53783336,474011756,
%T A355354 4231158016,38174676188,347566170384,3189295781780,29465038957708,
%U A355354 273851282010308,2558703740102840,24019990008557160,226444571054525156,2142925363606256584,20349477565111498148
%N A355354 G.f. A(x) satisfies: 4*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
%C A355354 a(n) = Sum_{k=0..n} A355350(n,k) * 4^k for n >= 0.
%H A355354 Paul D. Hanna, <a href="/A355354/b355354.txt">Table of n, a(n) for n = 0..400</a>
%F A355354 G.f. A(x) satisfies:
%F A355354 (1) 4*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
%F A355354 (2) 4*x*P(x) = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)), where P(x) = Product_{n>=1} 1/(1 - x^n) is the partition function (A000041), due to the Jacobi triple product identity.
%e A355354 G.f.: A(x) = 1 + 4*x + 28*x^2 + 196*x^3 + 1416*x^4 + 10860*x^5 + 87392*x^6 + 727188*x^7 + 6196212*x^8 + 53783336*x^9 + 474011756*x^10 + ...
%e A355354 where
%e A355354 4*x = ... - x^10/A(x)^5 + x^6/A(x)^4 - x^3/A(x)^3 + x/A(x)^2 - 1/A(x) + 1 - x*A(x) + x^3*A(x)^2 - x^6*A(x)^3 + x^10*A(x)^4 -+ ...
%e A355354 also,
%e A355354 4*x*P(x) = (1 - x*A(x))*(1 - 1/A(x)) * (1 - x^2*A(x))*(1 - x/A(x)) * (1 - x^3*A(x))*(1 - x^2/A(x)) * (1 - x^4*A(x))*(1 - x^3/A(x)) * ...
%e A355354 where P(x) is the partition function and begins
%e A355354 P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + 30*x^9 + 42*x^10 + 56*x^11 + 77*x^12 + ... + A000041(n)*x^n + ...
%o A355354 (PARI) {a(n) = my(A=[1,4],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(2*(#A)+9));
%o A355354 A[#A] = -polcoeff( sum(m=-t,t, (-1)^m*x^(m*(m+1)/2)*Ser(A)^m ), #A-1));A[n+1]}
%o A355354 for(n=0,30,print1(a(n),", "))
%Y A355354 Cf. A355350, A355351, A355352, A355353, A355355, A355356, A355357.
%K A355354 nonn
%O A355354 0,2
%A A355354 _Paul D. Hanna_, Jun 29 2022