A371984 Binomial transform of A371460.

1, 3, 15, 117, 1227, 16053, 251955, 4613997, 96566667, 2273672133, 59482039395, 1711735382877, 53737315411707, 1827584253650613, 66936582030410835, 2626714554845111757, 109948916113808074347, 4889877314768678051493

Offset: 0

Author

Prabha Sivaramannair, Apr 15 2024

Keywords

• nonn

Crossrefs

Cf. A371460.

Programs

Mathematica

nn = 17; a[0] = 1; Do[Set[a[n], 2^n + Sum[(3^j - 2^j)*Binomial[n, j]*a[n - j], {j, n}]], {n, nn}]; Array[a, nn + 1, 0] (* Michael De Vlieger, Apr 19 2024 *)

SageMath

def a(n):
    if n==0:
        return 1
    else:
        return 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^(2*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) = Sum_{j=1..n} (1-(-2)^j)*binomial(n,j)*a(n-j) for n > 0.

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

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