A318107 Triangle read by rows: T(n,k) = (3*n - 2*k)!/((n-k)!^3*k!).

1, 6, 1, 90, 24, 1, 1680, 630, 60, 1, 34650, 16800, 2520, 120, 1, 756756, 450450, 92400, 7560, 210, 1, 17153136, 12108096, 3153150, 369600, 18900, 336, 1, 399072960, 325909584, 102918816, 15765750, 1201200, 41580, 504, 1, 9465511770, 8779605120, 3259095840, 617512896, 63063000, 3363360, 83160, 720, 1

Offset: 0

Author

Gheorghe Coserea, Sep 18 2018

Keywords

• nonn
• tabl

Comments

Diagonal of rational function R(x,y,z,t) = 1/(1 - (x + y + z + t*x*y*z)) with respect to x,y,z, i.e., T(n,k) = [(xyz)^n*t^k] R(x,y,z,t).

Annihilating differential operator: x*(2*t*x + 1)*((t*x - 1)^3 + 27*x)*Dx^2 + (6*t^4*x^4 - 8*t^3*x^3 - 3*t*(t - 18)*x^2 + 6*(t + 9)*x - 1)*Dx + (t*x - 1)*(t*(2*t^2*x^2 + 2*t*x - 1) - 6).

Examples

			A(x;t) = 1 + (6 + t)*x + (90 + 24*t + t^2)*x^2 + (1680 + 630*t + 60*t^2 + t^3)*x^3 + ...
Triangle starts:
n\k [0]        [1]        [2]        [3]       [4]      [5]    [6]  [7]
[0] 1;
[1] 6,         1;
[2] 90,        24,        1;
[3] 1680,      630,       60,        1;
[4] 34650,     16800,     2520,      120,      1;
[5] 756756,    450450,    92400,     7560,     210,     1;
[6] 17153136,  12108096,  3153150,   369600,   18900,   336,   1;
[7] 399072960, 325909584, 102918816, 15765750, 1201200, 41580, 504, 1;
[8] ...
		

Links

• Gheorghe Coserea, Rows n=0..100, flattened

Crossrefs

Cf. A000172, A006480, A081798, A124435, A318108, A318109.

Programs

PARI

T(n,k) = (3*n - 2*k)!/((n-k)!^3*k!);
concat(vector(10, 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) = { \\ apply diffop p (encoded as Pol in Dx) to Ser 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; t='t;
diag(1/(1 - (x+y+z + t*x*y*z)), 11, [x,y,z]) == vector(11, n, P(n-1))
\\ annihilating diffop:
y = Ser(vector(101, n, P(n-1)), 'x);
p=x*(2*t*x + 1)*((t*x - 1)^3 + 27*x)*Dx^2 + (6*t^4*x^4 - 8*t^3*x^3 - 3*t*(t - 18)*x^2 + 6*(t + 9)*x  - 1)*Dx + (t*x - 1)*(t*(2*t^2*x^2 + 2*t*x - 1) - 6);
0 == apply_diffop(p, y)
*/

Formula

Let P_n(t) = Sum_{k=0..n} T(n,k)*t^k. Then A000172(n) = P_n(-4), A318108(n) = P_n(-3), A318109(n) = P_n(-2), A124435(n) = P_n(-1), A006480(n) = P_n(0), A081798(n) = P_n(1).

G.f. y = Sum_{n>=0} P_n(t)*x^n satisfies:

0 = x*(2*t*x + 1)*((t*x - 1)^3 + 27*x)*y'' + (6*t^4*x^4 - 8*t^3*x^3 - 3*t*(t - 18)*x^2 + 6*(t + 9)*x - 1)*y' + (t*x - 1)*(t*(2*t^2*x^2 + 2*t*x - 1) - 6)*y.