OFFSET
0,2
COMMENTS
Diagonal of rational function R(x,y,z,w,t) = 1/(1 - (x+y+z+w + t*x*y*z*w)) with respect to x,y,z,w, i.e., T(n,k) = [(xyzw)^n*t^k] R(x,y,z,w,t).
Annihilating differential operator: x^2*(3*t*x + 1)^2*((t*x - 1)^4 - 256*x)*Dx^3 + 3*x*(3*t*x + 1)*((t*x - 1)^3*(6*t^2*x^2 + 3*t*x - 1) - 384*x*(t*x + 1))*Dx^2 + (t*x - 1)*((t*x - 1)*(63*t^4*x^4 + 66*t^3*x^3 - 18*t*x + 1) + 48*x*(15*t*x + 17))*Dx + (t*x - 1)*(t*(9*t^4*x^4 + 12*t^3*x^3 + 6*t^2*x^2 - 12*t*x + 1) - 24*(15*t*x - 1)).
LINKS
Gheorghe Coserea, Rows n=0..100, flattened
FORMULA
EXAMPLE
A(x;t) = 1 + (24 + t)*x + (2520 + 120*t + t^2)*x^2 + (369600 + 22680*t + 360*t^2 + t^3)*x^3 + ...
Triangle starts:
n\k [0] [1] [2] [3] [4] [5] [6]
[0] 1;
[1] 24, 1;
[2] 2520, 120, 1;
[3] 369600, 22680, 360, 1;
[4] 63063000, 4804800, 113400, 840, 1;
[5] 11732745024, 1072071000, 33633600, 415800, 1680, 1;
[6] 2308743493056, 246387645504, 9648639000, 168168000, 1247400, 3024, 1;
[7] ...
MATHEMATICA
t[n_, k_] := (4*n - 3*k)!/((n-k)!^4*k!); Table[t[n, k], {n, 0, 10}, {k , 0, n}] // Flatten (* Amiram Eldar, Nov 07 2018 *)
PROG
(PARI) T(n, k) = (4*n-3*k)!/(k!*(n-k)!^4);
concat(vector(8, n, vector(n, k, T(n-1, k-1))))
/*
test:
P(n, v='t) = subst(Polrev(vector(n+1, k, T(n, k-1)), 't), 't, v);
diag(expr, N=22, var=variables(expr)) = {
my(a = vector(N));
for (k = 1, #var, expr = taylor(expr, var[#var - k + 1], N));
for (n = 1, N, a[n] = expr;
for (k = 1, #var, a[n] = polcoef(a[n], n-1)));
return(a);
};
apply_diffop(p, s) = {
s=intformal(s);
sum(n=0, poldegree(p, 'Dx), s=s'; polcoef(p, n, 'Dx) * s);
};
\\ diagonal property:
x='x; y='y; z='z; w='w; t='t;
diag(1/(1 - (x+y+z+w + t*x*y*z*w)), 9, [x, y, z, w]) == vector(9, n, P(n-1))
\\ annihilating diffop:
y = Ser(vector(101, n, P(n-1)), 'x);
p = x^2*(3*t*x + 1)^2*((t*x - 1)^4 - 256*x)*Dx^3 + 3*x*(3*t*x + 1)*((t*x - 1)^3*(6*t^2*x^2 + 3*t*x - 1) - 384*x*(t*x + 1))*Dx^2 + (t*x - 1)*((t*x - 1)*(63*t^4*x^4 + 66*t^3*x^3 - 18*t*x + 1) + 48*x*(15*t*x + 17))*Dx + (t*x - 1)*(t*(9*t^4*x^4 + 12*t^3*x^3 + 6*t^2*x^2 - 12*t*x + 1) - 24*(15*t*x - 1));
0 == apply_diffop(p, y)
*/
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gheorghe Coserea, Oct 15 2018
STATUS
approved