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A306809 Binomial transform of the continued fraction expansion of e. 1

%I #23 Apr 23 2020 04:20:21

%S 2,3,6,12,23,46,98,213,458,972,2051,4322,9098,19113,40054,83748,

%T 174767,364086,757298,1572861,3262242,6757500,13981019,28894090,

%U 59652314,123032913,253522382,521957844,1073741831,2207135966,4533576578,9305762469,19088743546,39131924268

%N Binomial transform of the continued fraction expansion of e.

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

%H Jackson Earles, Aaron Li, Adam Nelson, Marlo Terr, Sarah Arpin, and Ilia Mishev <a href="https://www.colorado.edu/math/binomial-transforms-sequences-spring-2019">Binomial Transforms of Sequences</a>, CU Boulder Experimental Math Lab, Spring 2019.

%F 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.

%F Conjectures from _Colin Barker_, Mar 12 2019: (Start)

%F 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).

%F 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.

%F (End)

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

%t 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 *)

%o (Sage)

%o def OEISbinomial_transform(N, seq):

%o BT = [seq[0]]

%o k = 1

%o while k< N:

%o next = 0

%o j = 0

%o while j <=k:

%o next = next + ((binomial(k,j))*seq[j])

%o j = j+1

%o BT.append(next)

%o k = k+1

%o return BT

%o econt = oeis('A003417')

%o OEISbinomial_transform(50,econt)

%Y Cf. A003417 (continued fraction for e).

%K cofr,nonn,easy

%O 0,1

%A _Sarah Arpin_, Mar 11 2019

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)