OFFSET
0,2
COMMENTS
Annihilating differential operator: x*(2*x+5)*(27*x^3+20*x^2+7*x-1)*Dx^2 + (162*x^4+620*x^3+314*x^2+70*x-5)*Dx + 48*x^3+220*x^2+80*x+10.
LINKS
Gheorghe Coserea, Table of n, a(n) for n = 0..310
A. Bostan, S. Boukraa, J.-M. Maillard, and J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015.
Steffen Eger, On the Number of Many-to-Many Alignments of N Sequences, arXiv:1511.00622 [math.CO], 2015.
FORMULA
G.f.: hypergeom([1/12, 5/12], [1], 1728*x^5*(27*x^3+20*x^2+7*x-1)/(8*x^2+8*x-1)^3)/(1-8*x-8*x^2)^(1/4).
0 = x*(2*x+5)*(27*x^3+20*x^2+7*x-1)*y'' + (162*x^4+620*x^3+314*x^2+70*x-5)*y' + (48*x^3+220*x^2+80*x+10)*y, where y is the g.f.
Recurrence: n^2*(29*n - 49)*a(n) = (203*n^3 - 546*n^2 + 393*n - 90)*a(n-1) + 20*(29*n^3 - 107*n^2 + 123*n - 42)*a(n-2) + 3*(3*n - 7)*(3*n - 5)*(29*n - 20)*a(n-3). - Vaclav Kotesovec, Jul 07 2016
a(n) = Sum_{k = 0..n} C(n,k)*C(n+k,k)*C(k,n-k) (apply Eger, Theorem 3 to the set of column vectors S = {[1,0,0], [1,1,0], [1,0,1], [0,1,1]}). - Peter Bala, Jan 26 2018
a(n) = [x^n] Legendre_P(n,1 + 2*x + 2*x^2). - Peter Bala, Dec 24 2020
From Peter Bala, Mar 16 2023: (Start)
(29*n - 49)*n^2*a(n) = (203*n^3 - 546*n^2 + 393*n - 90)*a(n-1) + 20*(29*n^3 - 107*n^2 + 123*n - 42)*a(n-2) + 3*(3*n - 7)*(29*n - 20)*(3*n - 5)*a(n-3).
Conjecture: the supercongruence a(n*p^r) == a(n*p^(r-1)) (mod p^(2*r)) holds for positive integers n and r and all primes p >= 5. (End)
MAPLE
seq(add(binomial(n, k)*binomial(n+k, k)*binomial(k, n-k), k = 0..n), n = 0..20); # Peter Bala, Jan 26 2018
PROG
(PARI)
my(x='x, y='y, z='z);
R = 1/(1 - x - x*y - x*z - y*z);
diag(n, expr, var) = {
my(a = vector(n));
for (i = 1, #var, expr = taylor(expr, var[#var - i + 1], n));
for (k = 1, n, a[k] = expr;
for (i = 1, #var, a[k] = polcoeff(a[k], k-1)));
return(a);
};
diag(10, R, [x, y, z])
(PARI) \\ system("wget http://www.jjj.de/pari/hypergeom.gpi");
read("hypergeom.gpi");
N = 20; x = 'x + O('x^N);
Vec(hypergeom([1/12, 5/12], [1], 1728*x^5*(27*x^3+20*x^2+7*x-1)/(8*x^2+8*x-1)^3, N)/(1-8*x-8*x^2)^(1/4))
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Gheorghe Coserea, Jul 06 2016
STATUS
approved