A306809 Binomial transform of the continued fraction expansion of e.

2, 3, 6, 12, 23, 46, 98, 213, 458, 972, 2051, 4322, 9098, 19113, 40054, 83748, 174767, 364086, 757298, 1572861, 3262242, 6757500, 13981019, 28894090, 59652314, 123032913, 253522382, 521957844, 1073741831, 2207135966, 4533576578, 9305762469, 19088743546, 39131924268

Offset: 0

Author

Sarah Arpin, Mar 11 2019

Keywords

• nonn
• easy

Comments

Taking this sequence as a continued fraction it seems to converge to 2.31601650488979...

Examples

			For n = 3, the a(3) = binomial(3,0)*2 + binomial(3,1)*1 + binomial(3,2)*2 + binomial(3,3)*1 = 12.
		

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

Cf. A003417 (continued fraction for e).

Programs

Mathematica

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

Sage

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

Formula

a(n) = Sum_{k=0..n} 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 - 11*x + 27*x^2 - 41*x^3 + 40*x^4 - 22*x^5 + 6*x^6) / ((1 - x)*(1 - 2*x)^2*(1 - x + x^2)^2).

a(n) = 7*a(n-1) - 21*a(n-2) + 37*a(n-3) - 43*a(n-4) + 33*a(n-5) - 16*a(n-6) + 4*a(n-7) for n>6.

(End)