A371460 Binomial transform of A355409.

1, 2, 10, 80, 838, 10952, 171910, 3148280, 65890198, 1551389192, 40586247910, 1167964662680, 36666464437558, 1247011549249832, 45672691012357510, 1792280373542404280, 75021202465129000918, 3336499249170658956872, 157116438405334017308710, 7809681380575733223237080, 408621675981135189773468278

Offset: 0

Author

Prabha Sivaramannair, Mar 24 2024

Keywords

• nonn

Crossrefs

Cf. A355409.

Programs

SageMath

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

SageMath

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

Formula

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

a(0) = 1, a(n) = 1 + Sum_{j=1..n} (3^j-2^j)*binomial(n,j)*a(n-j) for n > 0.

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