A370163 a(0) = 2, a(n) = (-1)^n + (-2)^n + (1/2) * Sum_{j=1..n} (1-(-1)^j-(-2)^j) * binomial(n,j) * a(n-j) for n > 0.

2, 1, 5, 25, 161, 1321, 13025, 149605, 1963841, 29004721, 475975745, 8591917885, 169193833121, 3609452038921, 82924458549665, 2041207822721365, 53594538159184001, 1495143168658285921, 44164021453758342785, 1377005070100813288045, 45193800193226286112481

Offset: 0

Author

Prabha Sivaramannair, Feb 26 2024

Keywords

• nonn

Comments

Inverse binomial transform of A370092 + A370456.

Crossrefs

Cf. A370092, A370456.

Programs

PARI

seq(n)={my(p=exp(x + O(x*x^n))); Vec(serlaplace(2*(1 + p)/(1 + p + p^2 - p^3)))} \\ Andrew Howroyd, Feb 26 2024

SageMath

def a(m):
    if m==0:
        return 2
    else:
        return (-1)^m+(-2)^m+1/2*sum([(1-(-2)^j-(-1)^j)*binomial(m,j)*a(m-j) for j in [1,..,m]])
list(a(m) for m in [0,..,20])

SageMath

f=2*(1+e^x)/(1+e^x+e^(2*x)-e^(3*x))
print([(diff(f,x,i)).subs(x=0) for i in [0,..,20]])

Formula

E.g.f.: 2*(1 + exp(x))/(1 + exp(x) + exp(2*x) - exp(3*x)).