A320949 G.f. A(x) satisfies: Sum_{n>=0} (1+x)^(n^2) / (1 + A(x))^(n+1) = 1.

1, 3, 21, 360, 9957, 374253, 17525043, 974215107, 62408814861, 4516276087767, 363954839947020, 32309808570484494, 3132592962836443359, 329373550369411338717, 37333941689722714789794, 4538664786712251729910968, 589148225684126151109083696, 81336502478379195711058521051, 11901242590779904151733826269744, 1839862821593669244253225972075464, 299668244116570657639682105216317053

Offset

0,2

Comments

Compare to the identity: Sum_{n>=0} G(x)^n / (1 + G(x))^(n+1) = 1, which holds for all G(x) such that G(0) is finite.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..60

Formula

a(n) = 0 (mod 3) for n > 0 (conjecture).

Example

G.f.: A(x) = 1 + 3*x + 21*x^2 + 360*x^3 + 9957*x^4 + 374253*x^5 + 17525043*x^6 + 974215107*x^7 + 62408814861*x^8 + 4516276087767*x^9 + 363954839947020*x^10 + ...
such that
1 = 1/(1 + A(x)) + (1+x)/(1 + A(x))^2 + (1+x)^4/(1 + A(x))^3 + (1+x)^9/(1 + A(x))^4 + (1+x)^16/(1 + A(x))^5 + (1+x)^25/(1 + A(x))^6 + (1+x)^36/(1 + A(x))^7 + ...
Also,
1 = 1/(1 + A(x)) + A(x)/(1 + A(x))^2 + A(x)^2/(1 + A(x))^3 + A(x)^3(1 + A(x))^4 + A(x)^4/(1 + A(x))^5 + A(x)^5/(1 + A(x))^6 + A(x)^6/(1 + A(x))^7 + ...
a(n) appears to be divisible by 3 for n > 0:
(A(x) - 1)/3 = x + 7*x^2 + 120*x^3 + 3319*x^4 + 124751*x^5 + 5841681*x^6 + 324738369*x^7 + 20802938287*x^8 + 1505425362589*x^9 + 121318279982340*x^10 + ...

Prog

(PARI)
\p120
{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = round(polcoeff(sum(n=0, 600, (1+x +x*O(x^#A))^(n^2) / (1 + Ser(A))^(n+1)*1. ), #A-1))); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Sequence in context: A052445 A351130 A186271 * A361056 A101389 A353321

Adjacent sequences: A320946 A320947 A320948 * A320950 A320951 A320952

Keyword

nonn

Author

Paul D. Hanna, Nov 13 2018

Status

approved