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%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
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