A245375 - OEIS

%I #3 Jul 19 2014 22:08:51

%S 1,4,20,112,736,5632,49024,474112,5017600,57597952,712597504,

%T 9446981632,133474877440,2000265674752,31666683510784,527786775150592,

%U 9233419259084800,169106747636580352,3234542505882025984,64473076850860490752,1336621867385969704960,28769619371258703511552

%N  G.f.: Sum_{n>=0} x^n / ( (1+x)^(n+1) * (1 - 4*(n+1)*x) ).

%F  G.f.: Sum_{n>=0} n! * (4*x)^n * (1+x)^n / Product_{k=1..n} (1 + 4*k*x).

%F a(n) = Sum_{k=0..floor(n/2)} Sum_{i=0..k} (-1)^i * binomial(k,i) * 4^(n-k) * (k-i+1)^(n-k).

%e  G.f.: A(x) = 1 + 4*x + 20*x^2 + 112*x^3 + 736*x^4 + 5632*x^5 + 49024*x^6 +...

%e where we have the following series identity:

%e A(x) = 1/((1+x)*(1-4*x)) + x/((1+x)^2*(1-8*x)) + x^2/((1+x)^3*(1-12*x))+ x^3/((1+x)^4*(1-16*x))+ x^4/((1+x)^5*(1-20*x)) + x^5/((1+x)^6*(1-24*x)) +...

%e is equal to

%e A(x) = 1 + 4*x*(1+x)/(1+4*x) + 2!*(4*x)^2*(1+x)^2/((1+4*x)*(1+8*x)) + 3!*(4*x)^3*(1+x)^3/((1+4*x)*(1+8*x)*(1+12*x)) + 4!*(4*x)^4*(1+x)^4/((1+4*x)*(1+8*x)*(1+12*x)*(1+16*x)) + 5!*(4*x)^5*(1+x)^5/((1+4*x)*(1+8*x)*(1+12*x)*(1+16*x)*(1+20*x)) +...

%o  (PARI) {a(n)=polcoeff( sum(m=0, n, x^m/((1+x)^(m+1)*(1 - 4*(m+1)*x) +x*O(x^n))), n)}

%o for(n=0, 30, print1(a(n), ", "))

%o (PARI) {a(n)=polcoeff( sum(m=0, n, 4^m*m!*x^m*(1+x)^m/prod(k=1, m, 1+4*k*x +x*O(x^n))), n)}

%o for(n=0, 30, print1(a(n), ", "))

%o (PARI) {a(n)=sum(k=0, floor(n/2), sum(i=0, k, (-1)^i*binomial(k, i)*(k-i+1)^(n-k)*4^(n-k)))}

%o for(n=0, 30, print1(a(n), ", "))

%Y  Cf. A229046, A245373, A245374, A245376.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jul 19 2014