cp's OEIS Frontend

A355355 G.f. A(x) satisfies: 5*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

%I A355355 #5 Jun 30 2022 10:40:16
%S A355355 1,5,40,320,2660,23455,216540,2064055,20137945,200134600,2019406895,
%T A355355 20635313325,213109960895,2220820915065,23323755734820,
%U A355355 246616999661690,2623193780773530,28049464032800110,301340494687086960,3251017466141039095,35207152686408604400
%N A355355 G.f. A(x) satisfies: 5*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
%C A355355 a(n) = Sum_{k=0..n} A355350(n,k) * 5^k for n >= 0.
%H A355355 Paul D. Hanna, <a href="/A355355/b355355.txt">Table of n, a(n) for n = 0..400</a>
%F A355355 G.f. A(x) satisfies:
%F A355355 (1) 5*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
%F A355355 (2) 5*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 A355355 G.f.: A(x) = 1 + 5*x + 40*x^2 + 320*x^3 + 2660*x^4 + 23455*x^5 + 216540*x^6 + 2064055*x^7 + 20137945*x^8 + 200134600*x^9 + 2019406895*x^10 + ...
%e A355355 where
%e A355355 5*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 A355355 also,
%e A355355 5*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 A355355 where P(x) is the partition function and begins
%e A355355 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 A355355 (PARI) {a(n) = my(A=[1,5],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(2*(#A)+9));
%o A355355 A[#A] = -polcoeff( sum(m=-t,t, (-1)^m*x^(m*(m+1)/2)*Ser(A)^m ), #A-1));A[n+1]}
%o A355355 for(n=0,30,print1(a(n),", "))
%Y A355355 Cf. A355350, A355351, A355352, A355353, A355354, A355356, A355357.
%K A355355 nonn
%O A355355 0,2
%A A355355 _Paul D. Hanna_, Jun 29 2022