A326558 G.f. A(x) satisifes: Sum_{n>=0} A(x)^((n-1)^2) * x^n = Sum_{n>=0} ((1+x)^(n-1) + 1)^n * x^n.

1, 1, 3, 10, 33, 126, 535, 2493, 12569, 68092, 394029, 2423257, 15767827, 108113333, 778198815, 5859990774, 46016573248, 375739918599, 3181928792057, 27881971943496, 252291917965337, 2353166240022595, 22588471117451579, 222844380918633385, 2256632985093301978, 23430724178641674376, 249198423317854694260, 2712372210346517891943

Offset

0,3

Links

Table of n, a(n) for n=0..27.

Formula

G.f. A(x) satisifes:

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

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

Example

G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 33*x^4 + 126*x^5 + 535*x^6 + 2493*x^7 + 12569*x^8 + 68092*x^9 + 394029*x^10 + 2423257*x^11 + 15767827*x^12 + ...
such that the following series are all equal
B(x) = A(x) + x + A(x)*x^2 + A(x)^4*x^3 + A(x)^9*x^4 + A(x)^16*x^5 + A(x)^25*x^6 + A(x)^36*x^7 + A(x)^49*x^8 + ... + A(x)^((n-1)^2) * x^n + ...
and
B(x) = 1 + (1 + 1)*x + (1 + (1+x))^2*x^2 + (1 + (1+x)^2)^3*x^3 + (1 + (1+x)^3)^4*x^4 + (1 + (1+x)^4)^5*x^5 + ... + (1 + (1+x)^(n-1))^n*x^n + ...
also
B(x) = 1/(1 - x) + x/(1 - x*(1+x))^2 + (1+x)^2*x^2/(1 - x*(1+x)^2)^3 + (1+x)^6*x^3/(1 - x*(1+x)^3)^4 + ... + (1+x)^(n*(n-1))*x^n/(1 - x*(1+x)^n)^(n+1) + ...
where
B(x) = 1 + 2*x + 4*x^2 + 12*x^3 + 41*x^4 + 164*x^5 + 728*x^6 + 3546*x^7 + 18679*x^8 + 105445*x^9 + 633198*x^10 + 4021124*x^11 + 26876020*x^12 + ...

Prog

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

Crossrefs

Cf. A326557.

Sequence in context: A149029 A149030 A024426 * A149031 A145928 A006535

Adjacent sequences: A326555 A326556 A326557 * A326559 A326560 A326561

Keyword

nonn

Author

Paul D. Hanna, Sep 16 2019

Status

approved