A333482 a(n) = [x^(2*n)] S(x)^n, where S(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) is the o.g.f. of the large Schröder numbers A006318.

1, 6, 304, 17718, 1093760, 69690006, 4530426640, 298634382374, 19886739416064, 1334658881073894, 90125657301992304, 6116315760393531094, 416791616968522726784, 28500344434239455360758, 1954614576511349850157392, 134392738169746273774331718, 9260873342398000417556078592

Offset

0,2

Comments

Compare with the sequence A103885(n) = [x^n] S(x)^n. See also A333481.

The Gauss congruences a(n*p^k) == a(n*p^(k-1)) ( mod p^k ) hold for all prime p and positive integers n and k.

We conjecture that the stronger supercongruences a(n*p^k) == a(n*p^(k-1)) ( mod p^(3*k) ) hold for prime p >= 5 and positive integers n and k.

More generally, we conjecture that for any positive integer a and any integer b, the sequence u(a,b;n) := [x^(a*n)] S(x)^(b*n) also satisfies the above congruences.

Links

Table of n, a(n) for n=0..16.

Formula

a(n) = (1/3) * Sum_{k = 0..2*n} C(3*n,k)*C(5*n-k-1,3*n-1) for n >= 1.

P-recursive: P(n)*s(n+1) = 2*(1023893*n^8 - 1278327*n^6 + 474507*n^4 - 57533*n^2 + 1620)*s(n) + P(-n)*s(n-1), where P(n) = 3*(n + 1)*(2*n + 1)*(3*n + 1)*(3*n + 2)*(533*n^4 - 1066*n^3 + 736*n^2 - 203*n + 18).

a(n) ~ (1921 + 533*sqrt(13))^n / (13^(1/4) * sqrt(Pi*n) * 2^(n+1) * 3^(3*n + 1/2)). - Vaclav Kotesovec, Mar 28 2020

Example

Examples of congruences:
a(17) - a(1) = 639400846289617183203551941830 - 6 = (2^6)*3*(17^3)* 677836910460361523003969 == 0 ( mod 17^3 ).
a(2*7) - a(2) = 1954614576511349850157392 - 304 = (2^5)*(7^4)*12408377* 2050236754217 == 0 ( mod 7^3 ).
a(5^2) - a(5) = 346904370487885277935868635823142219775940006 - 69690006 = (2^4)*(5^8)*911*39097145981*1558354721574649484551883 == 0 ( mod 5^6 )

Maple

[1, seq((1/3)*add(binomial(3*n, k)*binomial(5*n-k-1, 3*n-1), k = 0..3*n), n = 1..25)];
# alternative program
S := x -> (1/2)*(1-x-sqrt(1-6*x+x^2))/x:
G := (x, n) -> series(S(x)^n, x, 82):
seq(coeff(G(x, n), x, 2*n), n = 0..25);

Mathematica

Join[{1}, Table[Binomial[5*n-1, 3*n-1] * Hypergeometric2F1[-3*n, -2*n, 1 - 5*n, -1]/3, {n, 1, 20}]] (* Vaclav Kotesovec, Mar 28 2020 *)

Crossrefs

Cf. A006318, A103885, A333481.

Sequence in context: A159494 A221850 A229148 * A104003 A280215 A015103

Adjacent sequences: A333479 A333480 A333481 * A333483 A333484 A333485

Keyword

nonn,easy

Author

Peter Bala, Mar 24 2020

Status

approved