A318105 - OEIS

%I #20 Nov 09 2018 10:52:16

%S 1,24,1,2520,120,1,369600,22680,360,1,63063000,4804800,113400,840,1,

%T 11732745024,1072071000,33633600,415800,1680,1,2308743493056,

%U 246387645504,9648639000,168168000,1247400,3024,1,472518347558400,57718587326400,2710264100544,61108047000,672672000,3243240,5040,1

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

%C 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).

%C 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)).

%H Gheorghe Coserea, <a href="/A318105/b318105.txt">Rows n=0..100, flattened</a>

%F 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).

%e A(x;t) = 1 + (24 + t)*x + (2520 + 120*t + t^2)*x^2 + (369600 + 22680*t + 360*t^2 + t^3)*x^3 + ...

%e Triangle starts:

%e n\k [0]            [1]           [2]         [3]        [4]      [5]   [6]

%e [0] 1;

%e [1] 24,            1;

%e [2] 2520,          120,          1;

%e [3] 369600,        22680,        360,        1;

%e [4] 63063000,      4804800,      113400,     840,       1;

%e [5] 11732745024,   1072071000,   33633600,   415800,    1680,    1;

%e [6] 2308743493056, 246387645504, 9648639000, 168168000, 1247400, 3024, 1;

%e [7] ...

%t 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 *)

%o (PARI) T(n, k) = (4*n-3*k)!/(k!*(n-k)!^4);

%o concat(vector(8, n, vector(n, k, T(n-1, k-1))))

%o /*

%o test:

%o P(n, v='t) = subst(Polrev(vector(n+1, k, T(n, k-1)), 't), 't, v);

%o diag(expr, N=22, var=variables(expr)) = {

%o   my(a = vector(N));

%o   for (k = 1, #var, expr = taylor(expr, var[#var - k + 1], N));

%o   for (n = 1, N, a[n] = expr;

%o     for (k = 1, #var, a[n] = polcoef(a[n], n-1)));

%o   return(a);

%o };

%o apply_diffop(p, s) = {

%o   s=intformal(s);

%o   sum(n=0, poldegree(p, 'Dx), s=s'; polcoef(p, n, 'Dx) * s);

%o };

%o \\ diagonal property:

%o x='x; y='y; z='z; w='w; t='t;

%o diag(1/(1 - (x+y+z+w + t*x*y*z*w)), 9, [x, y, z, w]) == vector(9, n, P(n-1))

%o \\ annihilating diffop:

%o y = Ser(vector(101, n, P(n-1)), 'x);

%o 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));

%o 0 == apply_diffop(p, y)

%o */

%Y Cf. A008977, A082488, A125143, A318107.

%K nonn,tabl

%O 0,2

%A _Gheorghe Coserea_, Oct 15 2018