A110322 Row sums of A110321, a number triangle related to the Jacobsthal numbers.

1, 2, 9, 52, 425, 4206, 50737, 708464, 11350257, 204171130, 4084757561, 89849981772, 2156575777369, 56068679418662, 1569955094823585, 47098171778191816, 1507149193966389857, 51242941744764975474

Offset: 0

Author

Paul Barry, Jul 20 2005

Keywords

• easy
• nonn

Comments

Row sums of number triangle A110321.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A001045, A110321.

Programs

Mathematica

CoefficientList[Series[E^x/(1-x-2*x^2), {x, 0, 20}], x]* Table[n!, {n, 0, 20}] (* Vaclav Kotesovec, Oct 18 2012 *)

Formula

E.g.f.: exp(x)/(1-x-2*x^2).

a(n) = Sum_{k=0..n} n!*J(n-k+1)/k! where J(n)=A001045(n).

a(n) = Sum_{k=0..n} binomial(n, k)*k!*J(k+1) where J(n)=A001045(n).

a(n) ~ n!*2^(n+1)*exp(1/2)/3. - Vaclav Kotesovec, Oct 18 2012

Conjecture: a(n) +(-n-1)*a(n-1) -(2*n-1)*(n-1)*a(n-2) +2*(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Nov 11 2014

a(n) - n*a(n-1) - 2*n*(n-1)*a(n-2) - 1 = 0. - Martin Clever, Mar 22 2023

a(n) = (2*e^(1/2)*2^n*Gamma(n+1,1/2)+e^-1*(-1)^n*Gamma(n+1,-1))/3. - Martin Clever, Mar 25 2023