A318105 Triangle read by rows: T(n,k) = (4*n - 3*k)!/((n-k)!^4*k!).

1, 24, 1, 2520, 120, 1, 369600, 22680, 360, 1, 63063000, 4804800, 113400, 840, 1, 11732745024, 1072071000, 33633600, 415800, 1680, 1, 2308743493056, 246387645504, 9648639000, 168168000, 1247400, 3024, 1, 472518347558400, 57718587326400, 2710264100544, 61108047000, 672672000, 3243240, 5040, 1

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

Let P_n(t) = Sum_{k=0..n} T(n,k)*t^k. Then A125143(n) = P_n(-27), A008977(n) = P_n(0), A082488(n) = P_n(1).

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

Cf. A008977, A082488, A125143, A318107.

Sequence in context: A232988 A292781 A090215 * A040570 A040569 A040568

Adjacent sequences: A318102 A318103 A318104 * A318106 A318107 A318108

Keyword

nonn,tabl

Author

Gheorghe Coserea, Oct 15 2018

Status

approved