A326098 E.g.f.: Sum_{n>=0} (n+1) * (1 + x^n)^n * x^n/n!

1, 2, 7, 4, 77, 6, 2527, 8, 33609, 725770, 907211, 12, 946028173, 14, 968647695, 653837184016, 17473020364817, 18, 935267389056019, 20, 1723379337808128021, 1703031405723648022, 3716933623603223, 24, 124520749358323872153625, 3877802510832746496000026, 11787184960911360027, 10802449851605508096000028, 16938242101749730412851200029, 30, 51981534567681070815925862400031

Offset

0,2

Links

Table of n, a(n) for n=0..30.

Formula

E.g.f.: Sum_{n>=0} (n+1) * (1 + x^n)^n * x^n/n!.

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

a(n) = Sum_{d|n} (d + 1) * binomial(d, n/d - 1) * n!/d! for n>0, with a(0) = 1.

Example

E.g.f.: A(x) = 1 + 2*x + 7*x^2 + 4*x^3 + 77*x^4 + 6*x^5 + 2527*x^6 + 8*x^7 + 33609*x^8 + 725770*x^9 + 907211*x^10 + 12*x^11 + 946028173*x^12 + 14*x^13 + ...
such that
A(x) = 1 + 2*(1+x)*x + 3*(1+x^2)^2*x^2/2! + 4*(1+x^3)^3*x^3/3! + 5*(1+x^4)^4*x^4/4! + 6*(1+x^5)^5*x^5/5! + 7*(1+x^6)^6*x^6/6! + 8*(1+x^7)^7*x^7/7! + ...
also
A(x) = (1 + x)*exp(x) + (2 + x^2)*exp(x^2)*x^2 + (3 + x^3)*exp(x^3)*x^6/2! + (4 + x^4)*exp(x^4)*x^12/3! + (5 + x^5)*exp(x^5)*x^20/4! + (6 + x^6)*exp(x^6)*x^30/5! + (7 + x^7)*exp(x^7)*x^42/6! + ...

Prog

(PARI) {a(n) = if(n==0, 1, sumdiv(n, d, (d + 1) * binomial(d, n/d - 1) * n!/d! ) )}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* E.g.f.: Sum_{n>=0} (n+1) * (1 + x^n)^n * x^n/n! */
{a(n) = my(A = sum(m=0, n, (m+1) * (x^m + 1 +x*O(x^n))^m * x^m/m! )); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* E.g.f.: Sum_{n>=1} (n + x^n) * exp(x^n) * x^(n*(n-1))/(n-1)! */
{a(n) = my(A = sum(m=1, sqrtint(2*n+1), (m + x^m) * exp(x^m +x*O(x^n)) * x^(m*(m-1))/(m-1)! )); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A259208.

Sequence in context: A345273 A348145 A210421 * A198672 A306767 A113814

Adjacent sequences: A326095 A326096 A326097 * A326099 A326100 A326101

Keyword

nonn

Author

Paul D. Hanna, Jun 05 2019

Status

approved