A322729 G.f.: Sum_{n>=0} ((1+x)^n - 1 - x)^n / (2 - (1+x)^(n+1))^(n+1).

1, 1, 2, 20, 328, 6328, 143182, 3766776, 113468526, 3857336582, 146167530004, 6110935614644, 279466074847808, 13879201907633548, 743914773936904964, 42803931628586679640, 2631626471973012575402, 172174768825862821941366, 11943963245221499001128472, 875712464869008208365647312, 67663463933254286246310513580, 5495355957955926606903979413492

Offset

0,3

Links

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

Formula

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

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

Example

G.f.: A(x) = 1 + x + 2*x^2 + 20*x^3 + 328*x^4 + 6328*x^5 + 143182*x^6 + 3766776*x^7 + 113468526*x^8 + 3857336582*x^9 + 146167530004*x^10 + ...
such that
A(x) = 1/(2 - (1+x))  +  ((1+x) - 1-x)/(2 - (1+x)^2)^2  +  ((1+x)^2 - 1-x)^2/(2 - (1+x)^3)^3  +  ((1+x)^3 - 1-x)^3/(2 - (1+x)^4)^4  +  ((1+x)^4 - 1-x)^4/(2 - (1+x)^5)^5  +  ((1+x)^5 - 1-x)^5/(2 - (1+x)^6)^6 + ...
also
A(x) = 1/(2 + (1+x))  +  ((1+x) + 1+x)/(2 + (1+x)^2)^2  +  ((1+x)^2 + 1+x)^2/(2 + (1+x)^3)^3  +  ((1+x)^3 + 1+x)^3/(2 + (1+x)^4)^4  +  ((1+x)^4 + 1+x)^4/(2 + (1+x)^5)^5  +  ((1+x)^5 + 1+x)^5/(2 + (1+x)^6)^6 + ...

Prog

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

Crossrefs

Sequence in context: A375541 A323574 A294454 * A304861 A104462 A352601

Adjacent sequences: A322726 A322727 A322728 * A322730 A322731 A322732

Keyword

nonn

Author

Paul D. Hanna, Feb 02 2019

Status

approved