A337528 G.f.: Sum_{n>=0} (2^n + 1)^n * x^n / (1 + (2^n - 1)*x)^(n+1).

1, 3, 19, 513, 64447, 33221793, 68416620319, 561987558307353, 18435485678295497887, 2417353606284586529475393, 1267565977842177795997995695599, 2658400215575093543617417352025297273, 22300601642487504748989853483652483024500687

Offset

0,2

Comments

This sequence describes a specific case of the following identities:

(1) Sum_{n>=0} (q^n + p)^n * x^n / (1 + p*(q^n + p)*x)^(n+1) = Sum_{n>=0} (q^n - p)^n * x^n / (1 - p*(q^n - p)*x)^(n+1) ;

(2) Sum_{n>=0} (q^n + p)^n * exp(-p*(q^n + p)*x) * x^n/n! = Sum_{n>=0} (q^n - p)^n * exp( p*(q^n - p)*x) * x^n/n! ;

here, q = 2 and p = 1.

Links

Table of n, a(n) for n=0..12.

Formula

O.g.f.: Sum_{n>=0} (2^n + 1)^n * x^n / (1 + (2^n - 1)*x)^(n+1).

O.g.f.: Sum_{n>=0} (2^n - 1)^n * x^n / (1 - (2^n + 1)*x)^(n+1).

E.g.f.: Sum_{n>=0} (2^n + 1)^n * exp(-(2^n - 1)*x) * x^n/n!.

E.g.f.: Sum_{n>=0} (2^n - 1)^n * exp( (2^n + 1)*x) * x^n/n!.

a(n) = Sum_{k=0..n} binomial(n,k) * (2^k + 1)^(n-k) * (2^k - 1)^k.

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * (2^k - 1)^(n-k) * (2^k + 1)^k.

a(n) ~ 2^(n^2). - Vaclav Kotesovec, Aug 31 2020

Example

G.f.: A(x) = 1 + 3*x + 19*x^2 + 513*x^3 + 64447*x^4 + 33221793*x^5 + 68416620319*x^6 + 561987558307353*x^7 + 18435485678295497887*x^8 + 2417353606284586529475393*x^9 + 1267565977842177795997995695599*x^10 + ...
where
A(x) = 1 + 3*x/(1 + x)^2 + 5^2*x^2/(1 + 3*x)^3 + 9^3*x^3/(1 + 7*x)^4 + 17^4*x^4/(1 + 15*x)^5 + 33^5*x^5/(1 + 31*x)^6 + ...
Also, by a series identity,
A(x) = 1/(1 - 2*x) + x/(1 - 3*x)^2 + 3^2*x^2/(1 - 5*x)^3 + 7^3*x^3/(1 - 9*x)^4 + 15^4*x^4/(1 - 17*x)^5 + 31^5*x^5/(1 - 33*x)^6 + ...
EXPONENTIAL GENERATING FUNCTION.
E.g.f.: B(x) = 1 + 3*x + 19*x^2/2! + 513*x^3/3! + 64447*x^4/4! + 33221793*x^5/5! + 68416620319*x^6/6! + 561987558307353*x^7/7! + 18435485678295497887*x^8/8! + 2417353606284586529475393*x^9/9! + ...
where
B(x) = 1 + 3*exp(-x)*x + 5^2*exp(-3*x)*x^2/2! + 9^3*exp(-7*x)*x^3/3! + 17^4*exp(-15*x)*x^4/4! + 33^5*exp(-31*x)*x^5/5! + ...
Also, by a series identity,
B(x) = exp(2*x) + exp(3*x)*x + 3^2*exp(5*x)*x^2/2! + 7^3*exp(9*x)*x^3/3! + 15^4*exp(17*x)*x^4/4! + 31^5*exp(33*x)*x^5/5! + ...

Prog

(PARI) {a(n) = my(p=1, q=2, k=1);
A = sum(m=0, n, (q^m + p)^m * x^m / (1 - k*x + p*q^m*x +x*O(x^n))^(m+1) +x*O(x^n));
polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = my(p=1, q=2, k=1);
A = sum(m=0, n, (q^m - p)^m * x^m / (1 - k*x - p*q^m*x +x*O(x^n))^(m+1) +x*O(x^n));
polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = sum(k=0, n, binomial(n, k) * (2^k + 1)^(n-k) * (2^k - 1)^k )}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n, k) * (2^k - 1)^(n-k) * (2^k + 1)^k )}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A337527.

Sequence in context: A079306 A051381 A307080 * A136372 A272571 A355615

Adjacent sequences: A337525 A337526 A337527 * A337529 A337530 A337531

Keyword

nonn

Author

Paul D. Hanna, Aug 30 2020

Status

approved