A245376 G.f.: Sum_{n>=0} x^n / ( (1+x)^(n+1) * (1 - 5*(n+1)*x) ).

1, 5, 30, 200, 1550, 14000, 144500, 1662500, 20952500, 286437500, 4221312500, 66703437500, 1124194062500, 20109785937500, 380209901562500, 7571141773437500, 158312671414062500, 3466819503710937500, 79316483272226562500, 1891747084452148437500, 46942864023040039062500

Offset

0,2

Links

Table of n, a(n) for n=0..20.

Formula

G.f.: Sum_{n>=0} n! * (5*x)^n * (1+x)^n / Product_{k=1..n} (1 + 5*k*x).

a(n) = Sum_{k=0..floor(n/2)} Sum_{i=0..k} (-1)^i * binomial(k,i) * 5^(n-k) * (k-i+1)^(n-k).

Example

 G.f.: A(x) = 1 + 5*x + 30*x^2 + 200*x^3 + 1550*x^4 + 14000*x^5 +...
where we have the following series identity:
A(x) = 1/((1+x)*(1-5*x)) + x/((1+x)^2*(1-10*x)) + x^2/((1+x)^3*(1-15*x))+ x^3/((1+x)^4*(1-20*x))+ x^4/((1+x)^5*(1-25*x)) + x^5/((1+x)^6*(1-30*x)) +...
is equal to
A(x) = 1 + 5*x*(1+x)/(1+5*x) + 2!*(5*x)^2*(1+x)^2/((1+5*x)*(1+10*x)) + 3!*(5*x)^3*(1+x)^3/((1+5*x)*(1+10*x)*(1+15*x)) + 4!*(5*x)^4*(1+x)^4/((1+5*x)*(1+10*x)*(1+15*x)*(1+20*x)) + 5!*(5*x)^5*(1+x)^5/((1+5*x)*(1+10*x)*(1+15*x)*(1+20*x)*(1+25*x)) +...

Prog

 (PARI) {a(n)=polcoeff( sum(m=0, n, x^m/((1+x)^(m+1)*(1 - 5*(m+1)*x) +x*O(x^n))), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=polcoeff( sum(m=0, n, 5^m*m!*x^m*(1+x)^m/prod(k=1, m, 1+5*k*x +x*O(x^n))), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=sum(k=0, floor(n/2), sum(i=0, k, (-1)^i*binomial(k, i)*(k-i+1)^(n-k)*5^(n-k)))}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A229046, A245373, A245374, A245375.

Sequence in context: A107265 A196678 A128328 * A118346 A234422 A091927

Adjacent sequences: A245373 A245374 A245375 * A245377 A245378 A245379

Keyword

nonn

Author

Paul D. Hanna, Jul 19 2014

Status

approved