login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A243919 G.f.: Sum_{n>=0} (x^n - 1)^n * x^n / (1-x)^(n+1). 6

%I #10 Jul 02 2014 04:19:38

%S 1,0,1,2,1,-2,-3,1,9,13,6,-13,-33,-39,-18,33,102,159,158,46,-199,-525,

%T -781,-744,-221,808,2089,3101,3212,1933,-831,-4650,-8635,-11669,

%U -12652,-10615,-4635,6314,23181,45855,71241,91000,90501,51044,-43113,-193400,-374830,-529580,-573509

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

%C This sequence has a complex pattern of signs.

%H Paul D. Hanna, <a href="/A243919/b243919.txt">Table of n, a(n) for n = 0..1000</a>

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

%F G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (x^k - 1)^k.

%F G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (1 - x^k)^(n-k) * x^(k^2).

%e G.f.: A(x) = 1 + x^2 + 2*x^3 + x^4 - 2*x^5 - 3*x^6 + x^7 + 9*x^8 + 13*x^9 +...

%e where we have the series identity:

%e A(x) = 1/(1-x) + (x-1)*x/(1-x)^2 + (x^2-1)^2*x^2/(1-x)^3 + (x^3-1)^3*x^3/(1-x)^4 + (x^4-1)^4*x^4/(1-x)^5 +...+ (x^n-1)^n * x^n / (1-x)^(n+1) +...

%e A(x) = 1 + x^2/(1-x+x^2)^2 + x^6/(1-x+x^3)^3 + x^12/(1-x+x^4)^4 + x^20/(1-x+x^5)^5 + x^30/(1-x+x^6)^6 +...+ x^(n*(n+1)) / (1-x + x^(n+1))^(n+1) +...

%e as well as the binomial identity:

%e A(x) = 1 + x*(1 + (x-1)) + x^2*(1 + 2*(x-1) + (x^2-1)^2) + x^3*(1 + 3*(x-1) + 3*(x^2-1)^2 + (x^3-1)^3) + x^4*(1 + 4*(x-1) + 6*(x^2-1)^2 + 4*(x^3-1)^3 + (x^4-1)^4) + x^5*(1 + 5*(x-1) + 10*(x^2-1)^2 + 10*(x^3-1)^3 + 5*(x^4-1)^4 + (x^5-1)^5) +...+ x^n * Sum_{k=0..n} binomial(n,k) * (x^k-1)^k +...

%e A(x) = 1 + x*(0 + x) + x^2*(0 + 2*(1-x)*x + x^4) + x^3*(0 + 3*(1-x)^2*x + 3*(1-x^2)*x^4 + x^9) + x^4*(0 + 4*(1-x)^3*x + 6*(1-x^2)^2*x^4 + 4*(1-x^3)*x^9 + x^16) + x^5*(0 + 5*(1-x)^4*x + 10*(1-x^2)^3*x^4 + 10*(1-x^3)^2*x^9 + 5*(1-x^4)*x^16 + x^25) +...+ x^n * Sum_{k=0..n} binomial(n,k) * (1-x^k)^(n-k) * x^(k^2) +...

%o (PARI) {a(n)=local(A); A=sum(m=0, n, (x^m - 1)^m * x^m / (1-x +x*O(x^n) )^(m+1) ); polcoeff(A, n)}

%o for(n=0, 60, print1(a(n), ", "))

%o (PARI) {a(n)=local(A); A=sum(m=0, sqrtint(n+1), x^(m*(m+1)) / (1-x + x^(m+1) +x*O(x^n) )^(m+1) ); polcoeff(A, n)}

%o for(n=0, 60, print1(a(n), ", "))

%o (PARI) {a(n)=polcoeff(sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)*(x^k-1)^k) +x*O(x^n)), n)}

%o for(n=0, 40, print1(a(n), ", "))

%o (PARI) {a(n)=polcoeff(sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)*(1-x^k)^(m-k)*x^(k^2)) +x*O(x^n)), n)}

%o for(n=0, 40, print1(a(n), ", "))

%Y Cf. A243988.

%K sign

%O 0,4

%A _Paul D. Hanna_, Jun 18 2014

%E Name changed by _Paul D. Hanna_, Jul 02 2014

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 16:34 EDT 2024. Contains 371254 sequences. (Running on oeis4.)