The On-Line Encyclopedia of Integer Sequences (OEIS)

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A345697

Expansion of the e.g.f. sqrt(1 / (2*exp(x) - 2*x*exp(x) - 1)).
(history; published version)

#15 by Vaclav Kotesovec at Thu Jul 01 03:35:28 EDT 2021

STATUS

proposed

approved

#14 by Jon E. Schoenfield at Fri Jun 25 04:09:10 EDT 2021

STATUS

editing

proposed

Discussion

Tue Jun 29

11:54

Vaclav Kotesovec: Is the formula "a(p) == -1 (mod p)" proven or is it a conjecture?

13:48

Mélika Tebni: My example with p = 13 gives
  a (p) = Sum_ {k = 1..floor (p / 2)} A014307 (k) * A008306 (p, k).
A014307 (1) * A008306 (p, 1) == -1 (mod p), because A014307 (1) = 1 and A008306 (p, 1) = (p-1)!
For k> = 2, A008306 (p, k) == 0 (mod p), result a (p) == -1 (mod p).
I do not know how to demonstrate the formula with the linear combination a (p) = Sum ...

#13 by Jon E. Schoenfield at Fri Jun 25 04:08:53 EDT 2021

NAME

Expansion of the e.g.f. sqrt(1 / (2*exp(x) - 2*x*exp(x) - 1)).

#12 by Jon E. Schoenfield at Fri Jun 25 04:07:58 EDT 2021

EXAMPLE

sqrt(1/(2*exp(x)-2*x*exp(x)-1)) = 1 + x^2/2! + 2*x^3/3! + 12*x^4/4! + 64*x^5/5! + 485*x^6/6! + 4038*x^7/7! + 39991*x^8/8! + 441992*x^9/9! + ...

STATUS

proposed

editing

#11 by Vaclav Kotesovec at Fri Jun 25 02:15:58 EDT 2021

STATUS

editing

proposed

#10 by Vaclav Kotesovec at Fri Jun 25 02:14:38 EDT 2021

FORMULA

a(n) ~ sqrt(2*c) * n^n / ((1-c)^(n+1) * exp(n)), where c = -LambertW(-exp(-1)/2). - Vaclav Kotesovec, Jun 25 2021

STATUS

proposed

editing

#9 by Alois P. Heinz at Thu Jun 24 09:20:47 EDT 2021

STATUS

editing

proposed

#8 by Alois P. Heinz at Thu Jun 24 09:20:33 EDT 2021

FORMULA

a(0)=1, a(1)=0, a(n) = Sum_{k=1..floor(n/2)} A014307(k)*A008306(n,k) for n > = 1.

#7 by Alois P. Heinz at Thu Jun 24 09:18:15 EDT 2021

FORMULA

The eE.g.f. y(x) satisfies y' = x*exp(x)*y^3.

#6 by Alois P. Heinz at Thu Jun 24 09:16:08 EDT 2021

MAPLE

A014307 := proc(n) option remember; `if`(n=0, 1 , 1+add((-1+binomial(n, k))*A014307(k), k=1..n-1)) end:seq(A014307(n), n=0..25);

A008306 := proc(n, k): if k=1 then (n-1)! ; elif n<=2*k-1 then 0; else (n-1)*procname(n-1, k)+(n-1)*procname(n-2, k-1) ; end if; end proc: seq(seq(A008306(n, k), k=1..floor(n/2)), n=2..12);

a := n-> add((A014307(k)*A008306(n, k)), k=1..floor(n/2)):a(0):=1 ; seq(a(n), n=0..24);

seq(a(n), n=0..24);

# second program:

a := series(sqrt((1/(2*exp(x)-2*x*exp(x)-1))), x=0, 25): seq(n!*coeff(a, x, n), n=0..24);

seq(n!*coeff(a, x, n), n=0..24);

STATUS

proposed

editing