A333092 a(n) is the n-th order Taylor polynomial (centered at 0) of S(x)^(3*n) evaluated at x = 1, where S(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) is the o.g.f. of the Schröder numbers A006318.

1, 7, 109, 1951, 36993, 724007, 14457421, 292732671, 5987886081, 123440423047, 2560421160109, 53373725431583, 1117198199782785, 23465732683090471, 494330214846965389, 10440064992542621951, 220978578227187097601, 4686426367646858888711, 99559270036968523118317

Offset

0,2

Comments

The sequence satisfies the Gauss congruences: a(n*p^k) == a(n*p^(k-1)) ( mod p^k ) for all prime 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 prime p >= 5 and positive integers n and k. Examples of these congruences are given below.

More generally, for each integer m, we conjecture that the sequence a_m(n) defined as the n-th order Taylor polynomial of S(x)^(m*n) evaluated at x = 1 satisfies the same congruences. For cases, see A333090 (m = 1) and A333091 (m = 2). For similarly defined sequences see A333093 through A333097.

Links

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

Formula

a(n) = [x^n] ( (1 + x)*S^3(x/(1 + x)) )^n.

O.g.f.: ( 1 + x*f'(x)/f(x) )/( 1 - x*f(x) ), where f(x) = 1 + 6*x + 66*x^2 + 902*x^3 + 13794*x^4 + ... = (1/x) * series reversion of ( x/S^3(x) ).

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*sqrt(85 + 21*sqrt(17)) * (349 + 85*sqrt(17))^n / (68 * sqrt(Pi*n) * 2^(5*n)). - Vaclav Kotesovec, Mar 28 2020

Example

n-th order Taylor polynomial of S(x)^(3*n):
  n = 0: S(x)^0 = 1 + O(x)
  n = 1: S(x)^3 = 1 + 6*x + O(x^2)
  n = 2: S(x)^6 = 1 + 12*x + 96*x^2 + O(x^3)
  n = 3: S(x)^9 = 1 + 18*x + 198*x^2 + 1734*x^3 + O(x^4)
  n = 4: S(x)^12 = 1 + 24*x + 336*x^2 + 3608*x^3 + 33024*x^4 + O(x^5)
Setting x = 1 gives a(0) = 1, a(1) = 1 + 6 = 7, a(2) = 1 + 12 + 96 = 109, a(3) = 1 + 18 + 198 + 1734 = 1951 and a(4) = 1 + 24 + 336 + 3608 + 33024 = 36993.
The triangle of coefficients of the n-th order Taylor polynomial of S(x)^(2*n), n >= 0, in descending powers of x begins
                                           row sums
  n = 0 |     1                                1
  n = 1 |     6    1                           7
  n = 2 |    96   12    1                    109
  n = 3 |  1734  198   18   1               1951
  n = 4 | 33024 3608  336  24   1          36993
   ...
This is a Riordan array belonging to the Hitting time subgroup of the Riordan group. The first column sequence [1, 6, 96, 1734, 33024, 648006, ...]  = [x^n] S(x)^(3*n), and may also satisfy the above congruences.
Examples of congruences:
a(13) - a(1) = 23465732683090471 - 7 = (2^5)*(3^4)*(13^3)*83*911*54497 == 0 ( mod 13^3 ).
a(3*7) - a(3) = 962815680123979633351467303 - 1951 = (2^3)*(7^3)*29*41* 1832861*161008076794727  == 0 ( mod 7^3 ).
a(5^2) - a(5) = 201479167004032422703424646224007 - 724007 = (2^5)*(5^6)* 402958334008064845406849291 == 0 ( mod 5^6 ).

Maple

S:= x -> (1/2)*(1-x-sqrt(1-6*x+x^2))/x:
G := (x, n) -> series(S(x)^(3*n), x, 101):
seq(add(coeff(G(x, n), x, n-k), k = 0..n), n = 0..25);

Mathematica

Table[SeriesCoefficient[((1+x)*(1 - 3*x*(1+x) + (x^2 + x - 1)*Sqrt[1 - 4*x*(1+x)]) / (2*x^3))^n, {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 28 2020 *)

Crossrefs

Cf. A006318, A333090 through A333097.

Sequence in context: A202515 A096498 A123804 * A274668 A239848 A274787

Adjacent sequences: A333089 A333090 A333091 * A333093 A333094 A333095

Keyword

nonn,easy

Author

Peter Bala, Mar 22 2020

Status

approved