A357397 a(n) = coefficient of x^n, n >= 0, in A(x) such that: 0 = Sum_{n>=1} ((1+x)^n - A(x))^n / (1+x)^(n^2).

1, 1, 1, 5, 37, 367, 4463, 63797, 1043961, 19208815, 392278493, 8802891869, 215335062049, 5704017709585, 162695460126735, 4972552233126827, 162156046298476305, 5620675413587870585, 206382551428754263839, 8003189847508668434429

Offset

0,4

Comments

All terms appear to be odd.

Links

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

Example

G.f.: A(X) = 1 + x + x^2 + 5*x^3 + 37*x^4 + 367*x^5 + 4463*x^6 + 63797*x^7 + 1043961*x^8 + 19208815*x^9 + 392278493*x^10 + ...
where
0 = ((1+x) - A(x))/(1+x) + ((1+x)^2 - A(x))^2/(1+x)^4 + ((1+x)^3 - A(x))^3/(1+x)^9 + ((1+x)^4 - A(x))^4/(1+x)^16 + ((1+x)^5 - A(x))^5/(1+x)^25 + ... + ((1+x)^n - A(x))^n/(1+x)^(n^2) + ...
equivalently,
0 = (1 - A(x)/(1+x)) + (1 - A(x)/(1+x)^2)^2 + (1 - A(x)/(1+x)^3)^3 + (1 - A(x)/(1+x)^4)^4 + (1 - A(x)/(1+x)^5)^5 + ... + (1 - A(x)/(1+x)^n)^n + ...

Prog

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

Crossrefs

Cf. A357398.

Sequence in context: A084358 A050351 A129137 * A276232 A055869 A208231

Adjacent sequences: A357394 A357395 A357396 * A357398 A357399 A357400

Keyword

nonn

Author

Paul D. Hanna, Oct 20 2022

Status

approved