login
A306809
Binomial transform of the continued fraction expansion of e.
1
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
0,1
COMMENTS
Taking this sequence as a continued fraction it seems to converge to 2.31601650488979...
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.
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)
EXAMPLE
For n = 3, the a(3) = binomial(3,0)*2 + binomial(3,1)*1 + binomial(3,2)*2 + binomial(3,3)*1 = 12.
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 *)
PROG
(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)
CROSSREFS
Cf. A003417 (continued fraction for e).
Sequence in context: A293363 A326021 A164363 * A103341 A023675 A029996
KEYWORD
nonn,easy
AUTHOR
Sarah Arpin, Mar 11 2019
STATUS
approved