A245465 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k) * (1/(1-x)^k - 1)^k.

1, 1, 2, 4, 11, 35, 128, 523, 2329, 11206, 57685, 315515, 1824216, 11097706, 70771858, 471589169, 3274334755, 23630689143, 176882416706, 1370600471230, 10975020795140, 90675899684369, 771893276793888, 6761498234340104, 60874834962590159, 562694002401250455

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..200

Formula

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

G.f.: Sum_{n>=0} x^n * (1-x)^n / ((1-x)^(n+1) + x)^(n+1). - Paul D. Hanna, Jan 20 2015

Example

G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 11*x^4 + 35*x^5 + 128*x^6 + 523*x^7 +...
where we have the identity:
A(x) = 1 + x*(1 + (1/(1-x)-1))
+ x^2*(1 + 2*(1/(1-x)-1) + (1/(1-x)^2-1)^2)
+ x^3*(1 + 3*(1/(1-x)-1) + 3*(1/(1-x)^2-1)^2 + (1/(1-x)^3-1)^3)
+ x^4*(1 + 4*(1/(1-x)-1) + 6*(1/(1-x)^2-1)^2 + 4*(1/(1-x)^3-1)^3 + (1/(1-x)^4-1)^4)
+ x^5*(1 + 5*(1/(1-x)-1) + 10*(1/(1-x)^2-1)^2 + 10*(1/(1-x)^3-1)^3 + 5*(1/(1-x)^4-1)^4 + (1/(1-x)^5-1)^5) +...
is equal to
A(x) = 1 + x*(0 + 1/(1-x))
+ x^2*(0 + 2*(1-1/(1-x))*1/(1-x) + 1/(1-x)^4)
+ x^3*(0 + 3*(1-1/(1-x))^2*1/(1-x) + 3*(1-1/(1-x)^2)*1/(1-x)^4 + 1/(1-x)^9)
+ x^4*(0 + 4*(1-1/(1-x))^3*1/(1-x) + 6*(1-1/(1-x)^2)^2*1/(1-x)^4 + 4*(1-1/(1-x)^3)*1/(1-x)^9 + 1/(1-x)^16)
+ x^5*(0 + 5*(1-1/(1-x))^4*1/(1-x) + 6*(1-1/(1-x)^2)^3*1/(1-x)^4 + 4*(1-1/(1-x)^3)^2*1/(1-x)^9 + 5*(1-1/(1-x)^4)*1/(1-x)^16 + 1/(1-x)^25) +...
Also,
A(x) = 1 + x*(1-x)/((1-x)^2 + x)^2 + x^2*(1-x)^2/((1-x)^3 + x)^3 + x^3*(1-x)^3/((1-x)^4 + x)^4 + x^4*(1-x)^4/((1-x)^5 + x)^5 + x^5*(1-x)^5/((1-x)^6 + x)^6 + x^6*(1-x)^6/((1-x)^7 + x)^7 +...

Prog

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

Crossrefs

Cf. A245464.

Sequence in context: A107378 A186998 A243788 * A219861 A193058 A179379

Adjacent sequences: A245462 A245463 A245464 * A245466 A245467 A245468

Keyword

nonn

Author

Paul D. Hanna, Jul 23 2014

Status

approved