A323682 G.f.: Sum_{n>=0} x^n * ((1+x)^n + sqrt(2)*i)^n / (1 + sqrt(2)*i*x*(1+x)^n)^(n+1), where i^2 = -1.

1, 1, 2, 9, 28, 117, 547, 2671, 14258, 81335, 490525, 3124166, 20903467, 146330944, 1068670938, 8119799153, 64030850716, 522964211565, 4415538541339, 38476720288240, 345511658683542, 3192911107457165, 30327219314890373, 295738960383147498, 2957765009957920537, 30309760321885910842, 317966262017144158784

Offset

0,3

Comments

Note that the generating function expands into a power series in x with only real integer coefficients.

Links

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

Formula

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

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

Example

G.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 28*x^4 + 117*x^5 + 547*x^6 + 2671*x^7 + 14258*x^8 + 81335*x^9 + 490525*x^10 + 3124166*x^11 + 20903467*x^12 + ...
Let r = sqrt(2)*i, so that r^2 = -2, then
A(x) = 1/(1 + r*x) + x*((1+x) + r)/(1 + r*x*(1+x))^2 + x^2*((1+x)^2 + r)^2/(1 + r*x*(1+x)^2)^3 + x^3*((1+x)^3 + r)^3/(1 + r*x*(1+x)^3)^4 + x^4*((1+x)^4 + r)^4/(1 + r*x*(1+x)^4)^5 + x^5*((1+x)^5 + r)^5/(1 + r*x*(1+x)^5)^6 + ...
also,
A(x) = 1/(1 - r*x) + x*((1+x) - r)/(1 - r*x*(1+x))^2 + x^2*((1+x)^2 - r)^2/(1 - r*x*(1+x)^2)^3 + x^3*((1+x)^3 - r)^3/(1 - r*x*(1+x)^3)^4 + x^4*((1+x)^4 - r)^4/(1 - r*x*(1+x)^4)^5 + x^5*((1+x)^5 - r)^5/(1 - r*x*(1+x)^5)^6 + ...

Prog

(PARI) {a(n) = my(r = sqrt(2)*I, A = sum(m=0, n+1, x^m*((1+x +x*O(x^n))^m + r)^m/(1 + r*x*(1+x +x*O(x^n))^m)^(m+1) )); round(polcoeff(A, n))}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(r = sqrt(2)*I, A = sum(m=0, n+1, x^m*((1+x +x*O(x^n))^m - r)^m/(1 - r*x*(1+x +x*O(x^n))^m)^(m+1) )); round(polcoeff(A, n))}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A323681, A323683, A323684, A323685.

Sequence in context: A098518 A128239 A307400 * A086511 A291632 A328281

Adjacent sequences: A323679 A323680 A323681 * A323683 A323684 A323685

Keyword

nonn

Author

Paul D. Hanna, Feb 12 2019

Status

approved