A372215 a(n) is equal to the n-th order Taylor polynomial (centered at 0) of G(x)^n evaluated at x = 1, where G(x) = (1 - 3*x - sqrt(1 - 10*x + 9*x^2))/(2*x) is the g.f. of A082298.

1, 5, 65, 1013, 16897, 292005, 5157569, 92456341, 1675300865, 30604622405, 562675808065, 10398428960309, 192983418877441, 3594314403564773, 67146652988725697, 1257656071818605013, 23609209650223284225, 444081728926392461445, 8367715805572617168449

Offset

0,2

Comments

The sequence satisfies the Gauss congruences: a(n*p^k) == a(n*p^(k-1)) ( mod p^k ) for all primes p and positive integers n and k.

We conjecture that the sequence satisfies the stronger supercongruences a(n*p^k) == a(n*p^(k-1)) ( mod p^(3*k) ) for all primes p >= 5 and positive integers n and k. Examples of these supercongruences are given below.

More generally, for each integer m, we conjecture that the sequence {a_m(n) : n >= 0}, defined by setting a_m(n) = the n-th order Taylor polynomial of G(x)^(m*n) evaluated at x = 1, satisfies the same supercongruences.

Links

Table of n, a(n) for n=0..18.

Formula

a(n) = [x^n] ( (1 + x)*G(x/(1 + x)) )^n.

O.g.f.: ( 1 + x*F'(x)/F(x) )/( 1 - x*F(x) ), where F(x) = (1/x)*Revert( x/G(x) ) = = 1 + 4*x + 36*x^2 + 420*x^3 + 5572*x^4 + ....

Row sums of the Riordan array ( 1 + x*F'(x)/F(x), x*F(x) ) belonging to the Hitting time subgroup of the Riordan group.

a(n) ~ 3^(3*n/2 + 3/4) * (1 + sqrt(3))^(2*n-1) / (sqrt(Pi*n) * 2^(n+1)). - Vaclav Kotesovec, May 02 2024

Example

n-th order Taylor polynomial of G(x)^n:
  n = 0: G(x)^0 = 1 + O(x)
  n = 1: G(x)^1 = 1 + 4*x + O(x^2)
  n = 2: G(x)^2 = 1 + 8*x + 56*x^2 + O(x^3)
  n = 3: G(x)^3 = 1 + 12*x + 108*x^2 + 892 *x^3 + O(x^4)
  n = 4: G(x)^4 = 1 + 16*x + 176*x^2 + 1680*x^3 + 15024*x^4 + O(x^5)
Setting x = 1 gives a(0) = 1, a(1) = 1 + 4 = 5, a(2) = 1 + 8 + 56 = 65, a(3) = 1 + 12 +  108 + 892 = 1013 and a(4) = 1 + 16 + 176 + 1680 + 15024 = 16897.
The triangle of coefficients of the n-th order Taylor polynomial of G(x)^n, n >= 0, in descending powers of x begins
                                            row sums
  n = 0 |     1                                 1
  n = 1 |     4      1                          5
  n = 2 |    56      8     1                   65
  n = 3 |   892    108    12    1            1013
  n = 4 | 15024   1680   176   16   1       16897
   ...
This is a Riordan array belonging to the Hitting time subgroup of the Riordan group.
Examples of supercongruences:
a(13) - a(1) = 3594314403564773 - 5 = (2^5)*(3^3)*(13^3)*29*7643*8543 == 0 (mod 13^3).
a(2*7) - a(2) = 67146652988725697 - 65 = (2^7)*3*(7^4)*23*3943*803057 = 0 (mod 7^4).

Maple

G := x -> (1/2)*(1 - 3*x - sqrt(1 - 10*x + 9*x^2))/x:
H := (x, n) -> series(G(x)^n, x, 21):
seq(add(coeff(H(x, n), x, k), k = 0..n), n = 0..20);

Mathematica

Table[SeriesCoefficient[(2*(1 + x)^2/(1 - 2*x + Sqrt[1 - 8*x]))^n, {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 02 2024 *)

Prog

(PARI) G(x) = (1 - 3*x - sqrt(1 - 10*x + 9*x^2))/(2*x);
a(n) = my(x='x+O('x^(n+2))); subst(Pol(Vec(G(x)^n)), 'x, 1); \\ Michel Marcus, May 07 2024

Crossrefs

Cf. A082298, A333090, A333093, A372214.

Sequence in context: A103974 A233093 A208588 * A249930 A207262 A006278

Adjacent sequences: A372212 A372213 A372214 * A372216 A372217 A372218

Keyword

nonn,easy

Author

Peter Bala, Apr 23 2024

Status

approved