cp's OEIS Frontend

A306809 Binomial transform of the continued fraction expansion of e.

%I A306809 #26 Aug 10 2024 16:22:21
%S A306809 2,3,6,12,23,46,98,213,458,972,2051,4322,9098,19113,40054,83748,
%T A306809 174767,364086,757298,1572861,3262242,6757500,13981019,28894090,
%U A306809 59652314,123032913,253522382,521957844,1073741831,2207135966,4533576578,9305762469,19088743546,39131924268
%N A306809 Binomial transform of the continued fraction expansion of e.
%C A306809 Taking this sequence as a continued fraction it seems to converge to 2.31601650488979...
%H A306809 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 A306809 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 A306809 Conjectures from _Colin Barker_, Mar 12 2019: (Start)
%F A306809 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 A306809 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 A306809 (End)
%e A306809 For n = 3, the a(3) = binomial(3,0)*2 + binomial(3,1)*1 + binomial(3,2)*2 + binomial(3,3)*1 = 12.
%t A306809 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 A306809 (Sage)
%o A306809 def OEISbinomial_transform(N, seq):
%o A306809     BT = [seq[0]]
%o A306809     k = 1
%o A306809     while k< N:
%o A306809         next = 0
%o A306809         j = 0
%o A306809         while j <=k:
%o A306809             next = next + ((binomial(k,j))*seq[j])
%o A306809             j = j+1
%o A306809         BT.append(next)
%o A306809         k = k+1
%o A306809     return BT
%o A306809 econt = oeis('A003417')
%o A306809 OEISbinomial_transform(50,econt)
%Y A306809 Cf. A003417 (continued fraction for e).
%K A306809 nonn,easy
%O A306809 0,1
%A A306809 _Sarah Arpin_, Mar 11 2019