A268554 Diagonal of the rational function 1/((1 - w - u v) * (1 - x y - x z - y z)).
1, 36, 6300, 1552320, 445945500, 139815211536, 46384755633216, 16009450307136000, 5689533506261190300, 2067982222137781950000, 765185639177176836418800, 287266309673587605560908800, 109149488451384203661831720000
Offset: 0
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..200
- 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.
- R. Meštrović, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.
- Jacques-Arthur Weil, Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"
Programs
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Maple
A268554 := proc(n) 1/(1-w-u*v)/(1-x*y-x*z-y*z) ; coeftayl(%,x=0,n) ; coeftayl(%,y=0,n) ; coeftayl(%,z=0,n) ; coeftayl(%,u=0,n) ; coeftayl(%,v=0,n) ; coeftayl(%,w=0,n) ; end proc: seq(A268554(2*n),n=0..40) ; # R. J. Mathar, Mar 10 2016
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Mathematica
Table[(4*n)!*(3*n)!/((n!)^3*(2*n)!^2), {n, 0, 15}] (* Vaclav Kotesovec, Jul 01 2016 *) -
PARI
my(x1='x1, x2='x2, x3='x3, y1='y1, y2='y2, y3='y3); R = 1/((1 - y1 - y2*y3) * (1 - x1*x2 - x1*x3 - x2*x3)); 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(11, R, [x1,x2,x3,y1,y2,y3]) \\ Gheorghe Coserea, Jun 30 2016
Formula
Conjecture: n^3*(2*n-1)*a(n) -6*(4*n-1)*(3*n-1)*(3*n-2)*(4*n-3)*a(n-1)=0. - R. J. Mathar, Mar 10 2016
From Vaclav Kotesovec, Jul 01 2016: (Start)
a(n) = (4*n)! * (3*n)! / ((n!)^3 * (2*n)!^2).
a(n) ~ 2^(4*n - 3/2) * 3^(3*n + 1/2) / (Pi^(3/2) * n^(3/2)).
(End)
0 = (-x^2+432*x^4)*y'''' + (-5*x+4320*x^3)*y''' + (-4+10644*x^2)*y'' + 6012*x*y' + 288*y, where y = 1 + 36*x^2 + 6300*x^4 + ... is the g.f. - Gheorghe Coserea, Jul 03 2016
From Peter Bala, Oct 16 2024: (Start)
a(n) = 4 * Sum_{k = 0..2*n-1} (-1)^(n+k) * binomial(2*n-1, k) * binomial(4*n+k-1, k) * A108625(2*n, 2*n-k) for n >= 1 (verified using the MulZeil procedure in Doron Zeilberger's MultiZeilberger package). Cf. A002897.
a(n) = binomial(4*n, 2*n)*binomial(3*n, n)*binomial(2*n, n).
The supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(3*k)) hold for all primes p >= 5 and positive integers n and k (apply Meštrović, Section 6, equation 39).
a(n) = [x^(2*n)] (1 + x)^(4*n) * [x^n] (1 + x)^(3*n) * [x^n] (1 + x)^(2*n) = [x^n] F(x)^(36*n), where F(x) = 1 + x + 52*x^2 + 6919*x^3 + 1266837*x^4 + 275133604*x^5 + 66468858333*x^6 + 17272069128056*x^7 + 4732687104502730*x^8 + 1350192483617697301*x^9 + 397617338885817524186*x^10 + ... appears to have integer coefficients (checked up to O(x^500)).
Let E(x) = exp(Sum_{n >= 1} (1/36) *a(n)*x^n/n). Then E(x) = 1 + x + 88*x^2 + 14461*x^3 + 3115089*x^4 + 781116715*x^5 + 215898182457*x^6 + 63857605571783*x^7 + 19853845202113934*x^8 + 6413541401057933731*x^9 + 2135530251738770328084*x^10 + ... appears to have integer coefficients (checked up to O(x^500)).
a(n) = 36 * [x^n] ( x/series_reversion(E(x)) )^n.
For integer r and positive integer s, define sequences {u(n) : n >= 0} and {v(n) : n >= 0} by setting u(n) = [x^(s*n)] F(x)^(r*n) and v(n) = [x^(s*n)] E(x)^(r*n). We conjecture that both u(n) and v(n) satisfy the above supercongruences. (End)
Comments