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

%I A318107 #24 Sep 20 2018 21:15:02
%S A318107 1,6,1,90,24,1,1680,630,60,1,34650,16800,2520,120,1,756756,450450,
%T A318107 92400,7560,210,1,17153136,12108096,3153150,369600,18900,336,1,
%U A318107 399072960,325909584,102918816,15765750,1201200,41580,504,1,9465511770,8779605120,3259095840,617512896,63063000,3363360,83160,720,1
%N A318107 Triangle read by rows: T(n,k) = (3*n - 2*k)!/((n-k)!^3*k!).
%C A318107 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).
%C A318107 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).
%H A318107 Gheorghe Coserea, <a href="/A318107/b318107.txt">Rows n=0..100, flattened</a>
%F A318107 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).
%F A318107 G.f. y = Sum_{n>=0} P_n(t)*x^n satisfies:
%F A318107 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.
%e A318107 A(x;t) = 1 + (6 + t)*x + (90 + 24*t + t^2)*x^2 + (1680 + 630*t + 60*t^2 + t^3)*x^3 + ...
%e A318107 Triangle starts:
%e A318107 n\k [0]        [1]        [2]        [3]       [4]      [5]    [6]  [7]
%e A318107 [0] 1;
%e A318107 [1] 6,         1;
%e A318107 [2] 90,        24,        1;
%e A318107 [3] 1680,      630,       60,        1;
%e A318107 [4] 34650,     16800,     2520,      120,      1;
%e A318107 [5] 756756,    450450,    92400,     7560,     210,     1;
%e A318107 [6] 17153136,  12108096,  3153150,   369600,   18900,   336,   1;
%e A318107 [7] 399072960, 325909584, 102918816, 15765750, 1201200, 41580, 504, 1;
%e A318107 [8] ...
%o A318107 (PARI)
%o A318107 T(n,k) = (3*n - 2*k)!/((n-k)!^3*k!);
%o A318107 concat(vector(10, n, vector(n, k, T(n-1, k-1))))
%o A318107 /* test:
%o A318107 P(n, v='t) = subst(Polrev(vector(n+1, k, T(n, k-1)), 't), 't, v);
%o A318107 diag(expr, N=22, var=variables(expr)) = {
%o A318107   my(a = vector(N));
%o A318107   for (k = 1, #var, expr = taylor(expr, var[#var - k + 1], N));
%o A318107   for (n = 1, N, a[n] = expr;
%o A318107     for (k = 1, #var, a[n] = polcoef(a[n], n-1)));
%o A318107   return(a);
%o A318107 };
%o A318107 apply_diffop(p, s) = { \\ apply diffop p (encoded as Pol in Dx) to Ser s
%o A318107   s=intformal(s);
%o A318107   sum(n=0, poldegree(p, 'Dx), s=s'; polcoef(p, n, 'Dx) * s);
%o A318107 };
%o A318107 \\ diagonal property:
%o A318107 x='x; y='y; z='z; t='t;
%o A318107 diag(1/(1 - (x+y+z + t*x*y*z)), 11, [x,y,z]) == vector(11, n, P(n-1))
%o A318107 \\ annihilating diffop:
%o A318107 y = Ser(vector(101, n, P(n-1)), 'x);
%o A318107 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);
%o A318107 0 == apply_diffop(p, y)
%o A318107 */
%Y A318107 Cf. A000172, A006480, A081798, A124435, A318108, A318109.
%K A318107 nonn,tabl
%O A318107 0,2
%A A318107 _Gheorghe Coserea_, Sep 18 2018