A259223 E.g.f.: Sum_{n>=1} x^(n^2) * exp(2*x^n) / n!.

1, 4, 12, 44, 80, 912, 448, 41344, 62784, 2424320, 11264, 319358976, 53248, 11623886848, 435891701760, 1801685209088, 1114112, 1504049698308096, 4980736, 210465332463861760, 5676771352434180096, 792945839748153344, 96468992, 79367059219950565588992

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..455

Formula

E.g.f.: -exp(2) + Sum_{n>=0} (2 + x^n)^n / n!.

a(n) = Sum_{d|n} 2^(d-n/d) * binomial(d, n/d) * n!/d! for n>=1.

Example

E.g.f.: A(x) = x + 4*x^2/2! + 12*x^3/3! + 44*x^4/4! + 80*x^5/5! + 912*x^6/6! +...
where
A(x) = x*exp(2*x) + x^4*exp(2*x^2)/2! + x^9*exp(2*x^3)/3! + x^16*exp(2*x^4)/4! + x^25*exp(2*x^5)/5! + x^36*exp(2*x^6)/6! +...
also
A(x) = -exp(2) + 1 + (2+x) + (2+x^2)^2/2! + (2+x^3)^3/3! + (2+x^4)^4/4! + (2+x^5)^5/5! + (2+x^6)^6/6! +...

Mathematica

a[n_] := DivisorSum[n, 2^(#-n/#) Binomial[#, n/#] n!/#!& ]; Array[a, 30] (* Jean-François Alcover, Dec 18 2015 *)

Prog

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

Crossrefs

Cf. A259209, A265943, A265270.

Sequence in context: A149357 A149358 A149359 * A167402 A320643 A060897

Adjacent sequences: A259220 A259221 A259222 * A259224 A259225 A259226

Keyword

nonn

Author

Paul D. Hanna, Jun 21 2015

Status

approved