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A136525 a(n) = (2^n + 1)*(2^n + n + 1)^(n-1). 1

%I

%S 1,3,35,1296,157437,68809488,117274907815,816249936543744,

%T 23585997104539765625,2828012919296320973299968,

%U 1396969787088550953695654296875,2819773093146732354646026240000000000

%N a(n) = (2^n + 1)*(2^n + n + 1)^(n-1).

%F E.g.f.: A(x) = Sum_{n>=0} 2^(n^2) * W(2^n*x)^(n+1) * x^n/n! ; also, a(n)/n! = coefficient of x^n in W(x)^(2^n+1) where W(x) = LambertW(-x)/(-x).

%e E.g.f: A(x) = 1 + 3x + 35x^2/2! + 1296x^3/3! + 157437x^4/4! +...

%e A(x) = W(x) + 2*W(2x)^2*x + 2^4*W(4x)^3*x^2/2! + 2^9*W(8*x)^4*x^3/3! +...

%e W(x) = LambertW(-x)/(-x) = 1 + x + 3*x^2/2! + 4^2*x^3/3! +...+ (n+1)^(n-1)*x^n/n! +...

%e This is a special application of the following identity.

%e Let F(x) be a formal power series in x such that F(0)=1, then

%e Sum_{n>=0} m^n * F(q^n*x)^b * log( F(q^n*x) )^n / n! =

%e Sum_{n>=0} x^n * [y^n] F(y)^(m*q^n + b) .

%e The e.g.f. of this sequence is derived as follows.

%e Let F(x) = W(x) = LambertW(-x)/(-x);

%e since log( W(x) ) = x*W(x) and

%e since W(x)^m = Sum_{n>=0} m*(m+n)^(n-1)*x^n/n! then

%e Sum_{n>=0} m^n * q^(n^2) * W(q^n*x)^(b+n) * x^n/ n! =

%e Sum_{n>=0} (m*q^n + b) * (m*q^n + b + n)^(n-1) * x^n.

%o (PARI) a(n)=local(W=sum(k=0,n,(k+1)^(k-1)*x^k/k!)); n!*polcoeff( (W+x*O(x^n))^(2^n+1), n)

%o (PARI) /* As coefficient of x^n in Series: */ a(n)=local(W=sum(k=0,n,(k+1)^(k-1)*x^k/k!)); n!*polcoeff(sum(i=0,n,2^(i^2)*subst(W,x,2^i*x+x*O(x^n))^(i+1)*x^i/i!),n)

%Y Cf. A136524.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jan 03 2008

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Last modified May 25 00:59 EDT 2019. Contains 323534 sequences. (Running on oeis4.)