A323686 G.f.: Sum_{n>=0} x^n * ((1+x)^n + (-1)^n)^n / (1 - (-1)^n*x*(1+x)^n)^(n+1).

1, 1, 6, 19, 98, 483, 2713, 16183, 102982, 694083, 4922791, 36609764, 284389271, 2300588164, 19330419610, 168313262055, 1515639311976, 14089546025807, 134997931892389, 1331256937687764, 13493866107417156, 140422597267156783, 1498630308004089329, 16386210045999041610, 183397194510164874411, 2099282060320291913938, 24556888617117856139092

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..500

Formula

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

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

Example

G.f.: A(x) = 1 + x + 6*x^2 + 19*x^3 + 98*x^4 + 483*x^5 + 2713*x^6 + 16183*x^7 + 102982*x^8 + 694083*x^9 + 4922791*x^10 + 36609764*x^11 + ...
such that
A(x) = 1/(1 - x) + x*((1+x) - 1)/(1 + x*(1+x))^2 + x^2*((1+x)^2 + 1)^2/(1 - x*(1+x)^2)^3 + x^3*((1+x)^3 - 1)^3/(1 + x*(1+x)^3)^4 + x^4*((1+x)^4 + 1)^4/(1 - x*(1+x)^4)^5 + x^5*((1+x)^5 - 1)^5/(1 + x*(1+x)^5)^6 + ...
also,
A(x) = 1/(1 + x) + x*((1+x) + 1)/(1 - x*(1+x))^2 + x^2*((1+x)^2 - 1)^2/(1 + x*(1+x)^2)^3 + x^3*((1+x)^3 + 1)^3/(1 - x*(1+x)^3)^4 + x^4*((1+x)^4 - 1)^4/(1 + x*(1+x)^4)^5 + x^5*((1+x)^5 + 1)^5/(1 - x*(1+x)^5)^6 + ...

Prog

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

Crossrefs

Cf. A323680.

Sequence in context: A151277 A192368 A355539 * A285853 A138748 A097899

Adjacent sequences: A323683 A323684 A323685 * A323687 A323688 A323689

Keyword

nonn

Author

Paul D. Hanna, Feb 12 2019

Status

approved