A267220 Expansion of exp( Sum_{n >= 1} A005259(n)*x^n/n ).

1, 5, 49, 685, 11807, 232771, 5031415, 116222699, 2822056474, 71230971002, 1854334597966, 49503739725470, 1349449053997654, 37438177610268014, 1054371288632733022, 30081609844254013942, 867990837575171240525, 25295504900322451251793

Offset

0,2

Comments

The expansion of exp( Sum_{n >= 1} A005259(n-1)*x^n/n ) may also have only integer coefficients. See the Example section.

This is proved in Beukers, p. 143. - Peter Bala, Mar 06 2020

Links

Table of n, a(n) for n=0..17.

F. Beukers, Some congruences for the Apery numbers, Journal of Number Theory, Vol. 21, Issue 2, Oct. 1985, pp. 141-155.

P. Peart and W.-J. Woan, A divisibility property for a subgroup of Riordan matrices, Discrete Applied Mathematics, Vol. 98, Issue 3, Jan 2000, 255-263.

Formula

n*a(n) = Sum_{k = 0..n-1} A005259(n-k)*a(k).

O.g.f.: A(x) = exp( Sum_{n >= 1} A005259(n)*x^n/n ) = 1 + 5*x + 49*x^2 + 685*x^3 + 11807*x^4 + 232771*x^5 + 5031415*x^6 + ....

The o.g.f. A(x) satisfies 1 + x* d/dx(log(A(x)) = Sum_{n >= 0} A005259(n)*x^n.

Let F(x) = x/( series reversion of x*A(x) ) = 1 + 5*x + 24*x^2 + 200*x^3 + 2430*x^4 + 36096*x^5 + 605620*x^6 + 11024496*x^7 + 212758245*x^8 + .... Then A005259(n) = [x^n]( F(x)^n ). See the example section.

Example

exp( Sum_{n >= 1} A005259(n-1)*x^n/n ) = 1 + 3*x + 27*x^2 + 390*x^3 + .... The coefficient list begins [1, 3, 27, 390, 7038, 144550, 3232294, 76768575, 1907248655, 49067872253, 1298071849821, 35131589549434, 969031280176162, 27161049933788970, 771837331889465586, 22196147941000214583, 644991902540171273463, ...].
The Apéry number sequence A005259 begins [1, 5, 73, 1445, 33001, 819005, ...]. Let F(x) = x/( series reversion of x*A(x) ) = 1 + 5*x + 24*x^2 + 200*x^3 + 2430*x^4 + ....
Truncation of F(x)^n:
F(x)^0: 1
F(x)^1: 1 + 5*x
F(x)^2: 1 + 10*x + 73*x^2
F(x)^3: 1 + 15*x + 147*x^2 + 1445*x^3
F(x)^4: 1 + 20*x + 246*x^2 + 2740*x^3 + 33001*x^4
The coefficient array
      1
      5    1
     73   10   1
   1445  147  15  1
  33001 2740 246 20 1
  ...
has the Apéry numbers as the first column.
It is a Riordan array belonging to the hitting-time subgroup of the Riordan group - see Peart and Woan.

Maple

#A267220
#define the Apéry numbers
A005259 := proc (n) option remember; if n = 0 then 1 elif n = 1 then 5 else ((34*(n-1)^3+51*(n-1)^2+27*n-22)*A005259(n-1)-(n-1)^3*A005259(n-2))/n^3 end if; end proc:
exp(add(A005259(n)*x^n/n, n = 1 .. 17)):
seq(coeftayl(%, x = 0, n), n = 0 .. 17);

Crossrefs

Cf. A005259, A267219.

Sequence in context: A116873 A324361 A089914 * A052142 A136729 A102773

Adjacent sequences: A267217 A267218 A267219 * A267221 A267222 A267223

Keyword

nonn,easy

Author

Peter Bala, Jan 12 2016

Status

approved