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A134174 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(2^k,n). 4
1, 1, 4, 44, 1546, 180096, 69656776, 90247564712, 398349418563148, 6107204301234263072, 331155704662551903747856, 64494012244416274972130226976, 45679970430008422082570864807000804, 118841746271330792971612314663523090972880, 1144692720783363684402532942114790915481788335224 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..59

FORMULA

G.f.: Sum_{n>=0} log(1 + (2^n-1)*x)^n / n!.

a(n) = (1/n!) * Sum_{k=0..n} Stirling1(n,k) * (2^k-1)^n.

From Vaclav Kotesovec, Jul 02 2016: (Start)

a(n) ~ binomial(2^n,n).

a(n) ~ 2^(n^2) / n!.

a(n) ~ 2^(n^2 - 1/2) * exp(n) / (sqrt(Pi) * n^(n+1/2)).

(End)

EXAMPLE

G.f.: A(x) = 1 + x + 4*x^2 + 44*x^3 + 1546*x^4 + 180096*x^5 +...

where

A(x) = 1 + log(1+x) + log(1+3*x)^2/2! + log(1+7*x)^3/3! + log(1+15*x)^4/4! + log(1+31*x)^5/5! + log(1+63*x)^6/6! + log(1+127*x)^7/7! + log(1+255*x)^8/8! +...

MATHEMATICA

Table[Sum[(-1)^(n-k)*Binomial[n, k]*Binomial[2^k, n], {k, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, Jul 02 2016 *)

PROG

(PARI) {a(n)=local(A=1, X=x+x*O(x^n)); A=sum(k=0, n, log(1+(2^k-1)*X)^k/k!); polcoeff(A, n)}

for(n=0, 20, print1(a(n), ", "))

(PARI) {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}

{a(n)=1/n!*sum(k=0, n, Stirling1(n, k)*(2^k-1)^n)}

for(n=0, 20, print1(a(n), ", "))

CROSSREFS

Cf. A133990.

Sequence in context: A306372 A217473 A127635 * A296729 A157193 A224784

Adjacent sequences:  A134171 A134172 A134173 * A134175 A134176 A134177

KEYWORD

easy,nonn

AUTHOR

Paul Hanna and Vladeta Jovovic, Jan 13 2008

STATUS

approved

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Last modified October 17 07:39 EDT 2021. Contains 348048 sequences. (Running on oeis4.)