A333481 a(n) = [x^n] S(x)^(2*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, 4, 48, 652, 9344, 138004, 2077968, 31712412, 488793088, 7591462564, 118615816048, 1862444310060, 29361743698304, 464472032918196, 7368841675386960, 117200150284494652, 1868129273410953216, 29834667873867329348, 477283242733227391152, 7647021589988643092428

Offset

0,2

Comments

Compare with the sequence A103885(n) = [x^n] S(x)^n, where S(x) is the o.g.f. of the large Schröder numbers A006318. See also A333482.

The Gauss congruences a(n*p^k) == a(n*p^(k-1)) ( mod p^k ) hold for 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..19.

Formula

a(n) = (2/3) * Sum_{k = 0..n} C(3*n,k)*C(4*n-k-1,3*n-1) for n >= 1.

P-recursive: P(n)*a(n + 1) = 4*(7805*n^6 - 7132*n^4 + 1559*n^2 - 72)*a(n) -

P(-n)*a(n - 1), where P(n) =(2*n - 1)*(3*n + 1)*(3*n + 2)*(3*n + 3)*(35*n^2 - 35*n + 6).

a(n) ~ 2^(1/4) * (223 + 70*sqrt(10))^n / (5^(1/4) * sqrt(Pi*n) * 3^(3*n + 1/2)). - Vaclav Kotesovec, Mar 28 2020

Example

Examples of congruences:
a(17) - a(1) = 29834667873867329348 - 4 = (2^6)*(17^3)*401*236619262717 == 0 ( mod 17^3 ).
a(2*7) - a(2) = 7368841675386960 - 48 = (2^5)*3*(7^4)*577*3229*17159 == 0 ( mod 7^3 ).
a(5^2) - a(5) = 132585158051380247023537388004 - 138004 = (2^4)*(5^6)* 39461*13439614612035199009 == 0 ( mod 5^6 )

Maple

[1, seq((2/3)*add(binomial(3*n, k)*binomial(4*n-k-1, 3*n-1), k = 0..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)^(2*n), x, 76):
seq(coeff(G(x, n), x, n), n = 0..25);

Mathematica

Join[{1}, Table[2*Binomial[4*n-1, 3*n-1] * Hypergeometric2F1[-3*n, -n, 1 - 4*n, -1]/3, {n, 1, 20}]] (* Vaclav Kotesovec, Mar 28 2020 *)

Crossrefs

Cf. A006318, A103885, A333482.

Sequence in context: A226705 A126967 A098402 * A003774 A214819 A211198

Adjacent sequences: A333478 A333479 A333480 * A333482 A333483 A333484

Keyword

nonn,easy

Author

Peter Bala, Mar 24 2020

Status

approved