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
Examples
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] ...
Links
- Gheorghe Coserea, Rows n=0..100, flattened
Programs
-
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 *) -
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) */
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