A325296 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.

1, 2, 6, 34, 274, 2566, 26406, 290530, 3361042, 40463894, 503505542, 6445263858, 84593906962, 1135730543782, 15571171913958, 217755224972034, 3103675765823634, 45064501714445366, 666402338952126790, 10035910959863435794, 153933449475479903634, 2405188381726250188486, 38293058095081812664742, 621408387360835449163042, 10281437987942851628839442, 173489555489829641553617494

Offset

0,2

Comments

a(n) = 2 (mod 4) for n > 0.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Formula

G.f. A(x) allows the following sums to equal the same series B(x):

(1) B(x) = Sum_{n>=0} x^n * (1 + A(x)^n)^n,

(2) B(x) = Sum_{n>=0} x^n * A(x)^(n^2) / (1 - x*A(x)^n)^(n+1).

(3) B(x) = 1 + 2*Sum_{n>=1} x^n * A(x)^(n^2).

FORMULAS FOR TERMS.

a(n) = 2 (mod 4) for n > 0.

Example

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 + ...
such that the following sums are all equal:
(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 + ...
(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 + ...
(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 + ...
where
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 + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = -polcoeff( sum(n=0, #A, x^n*(2*Ser(A)^(n^2) - (1+Ser(A)^n)^n) ), #A) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A326275, A326560, A326561, A326562, A326287.

Sequence in context: A002685 A262391 A271212 * A052878 A168362 A274711

Adjacent sequences: A325293 A325294 A325295 * A325297 A325298 A325299

Keyword

nonn

Author

Paul D. Hanna, Apr 23 2019

Status

approved