A306810 Inverse binomial transform of the continued fraction expansion of e.

2, -1, 2, -4, 7, -8, -2, 41, -134, 296, -485, 512, 82, -2107, 6562, -13852, 21871, -22600, -2186, 83105, -255878, 531440, -826685, 846368, 59050, -2952451, 9034498, -18600436, 28697815, -29229256, -1594322, 98848025, -301327046, 617003000, -947027861, 961376768, 43046722

Offset: 0

Author

Sarah Arpin, Mar 11 2019

Keywords

• easy
• sign

Examples

			For n = 3, a(3) = -binomial(3,0)*2 + binomial(3,1)*1 - binomial(3,2)*2 + binomial(3,3)*1 = -4.
		

Links

• Jackson Earles, Aaron Li, Adam Nelson, Marlo Terr, Sarah Arpin, and Ilia Mishev Binomial Transforms of Sequences, CU Boulder Experimental Math Lab, Spring 2019.

Crossrefs

Continued fraction of e: A003417.

Binomial transform of continued fraction of e: A306809.

Programs

Mathematica

nmax = 50; A003417 = ContinuedFraction[E, nmax+1]; Table[Sum[(-1)^(n + k)*Binomial[n, k]*A003417[[k + 1]], {k, 0, n}], {n, 0, nmax}] (* Vaclav Kotesovec, Apr 23 2020 *)

Sage

def OEISInverse(N, seq):
    BT = [seq[0]]
    k = 1
    while k< N:
        next = 0
        j = 0
        while j <=k:
            next = next + (((-1)^(j+k))*(binomial(k,j))*seq[j])
            j = j+1
        BT.append(next)
        k = k+1
    return BT
econt = oeis('A003417')
OEISInverse(50,econt)

Formula

a(n) = Sum{k=0...n}(-1)^(n+k)*binomial(n,k)*b(k), where b(k) is the k-th term of the continued fraction expansion of e.

Conjectures from Colin Barker, Mar 12 2019: (Start)

G.f.: (2 + 13*x + 37*x^2 + 55*x^3 + 42*x^4 + 14*x^5 + 2*x^6) / ((1 + x)*(1 + 3*x + 3*x^2)^2).

a(n) = - 7*a(n-1) - 21*a(n-2) - 33*a(n-3) - 27*a(n-4) - 9*a(n-5) for n>6.

(End)