|
%I #20 Jul 24 2019 20:07:46
%S 1,2,6,34,274,2566,26406,290530,3361042,40463894,503505542,6445263858,
%T 84593906962,1135730543782,15571171913958,217755224972034,
%U 3103675765823634,45064501714445366,666402338952126790,10035910959863435794,153933449475479903634,2405188381726250188486,38293058095081812664742,621408387360835449163042,10281437987942851628839442,173489555489829641553617494
%N G.f. A(x) satisfies: 1 + 2*Sum_{n>=1} x^n*A(x)^(n^2) = Sum_{n>=0} x^n*(1 + A(x)^n)^n.
%C a(n) = 2 (mod 4) for n > 0.
%H Paul D. Hanna, <a href="/A325296/b325296.txt">Table of n, a(n) for n = 0..300</a>
%F G.f. A(x) allows the following sums to equal the same series B(x):
%F (1) B(x) = Sum_{n>=0} x^n * (1 + A(x)^n)^n,
%F (2) B(x) = Sum_{n>=0} x^n * A(x)^(n^2) / (1 - x*A(x)^n)^(n+1).
%F (3) B(x) = 1 + 2*Sum_{n>=1} x^n * A(x)^(n^2).
%F FORMULAS FOR TERMS.
%F a(n) = 2 (mod 4) for n > 0.
%e G.f.: A(x) = 1 + 2*x + 6*x^2 + 34*x^3 + 274*x^4 + 2566*x^5 + 26406*x^6 + 290530*x^7 + 3361042*x^8 + 40463894*x^9 + 503505542*x^10 + 6445263858*x^11 + 84593906962*x^12 + 1135730543782*x^13 + 15571171913958*x^14 + 217755224972034*x^15 + 3103675765823634*x^16 + ...
%e such that the following sums are all equal:
%e (1) B(x) = 1 + x*(1 + A(x)) + x^2*(1 + A(x)^2)^2 + x^3*(1 + A(x)^3)^3 + x^4*(1 + A(x)^4)^4 + x^5*(1 + A(x)^5)^5 + x^6*(1 + A(x)^6)^6 + x^7*(1 + A(x)^7)^7 + x^8*(1 + A(x)^8)^8 + ...
%e (2) B(x) = 1/(1-x) + x*A(x)/(1-x*A(x))^2 + x^2*A(x)^4/(1-x*A(x)^2)^3 + x^3*A(x)^9/(1-x*A(x)^3)^4 + x^4*A(x)^16/(1-x*A(x)^4)^5 + x^5*A(x)^25/(1-x*A(x)^5)^6 + x^6*A(x)^36/(1-x*A(x)^6)^7 + x^7*A(x)^49/(1-x*A(x)^7)^8 + ...
%e (3) B(x) = 1 + 2*x*A(x) + 2*x^2*A(x)^4 + 2*x^3*A(x)^9 + 2*x^4*A(x)^16 + 2*x^5*A(x)^25 + 2*x^6*A(x)^36 + 2*x^7*A(x)^49 + 2*x^8*A(x)^64 + ...
%e where
%e B(x) = 1 + 2*x + 6*x^2 + 30*x^3 + 202*x^4 + 1634*x^5 + 14934*x^6 + 148862*x^7 + 1583578*x^8 + 17724802*x^9 + 206742342*x^10 + 2496080542*x^11 + 31043750570*x^12 + 396327038050*x^13 + 5180639658102*x^14 + 69207202312318*x^15 + 943572290565690*x^16 + ...
%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
%o A[#A] = -polcoeff( sum(n=0,#A, x^n*(2*Ser(A)^(n^2) - (1+Ser(A)^n)^n) ),#A) );A[n+1]}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A326275, A326560, A326561, A326562, A326287.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Apr 23 2019
| 