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A318107 Triangle read by rows: T(n,k) = (3*n - 2*k)!/((n-k)!^3*k!). 4
1, 6, 1, 90, 24, 1, 1680, 630, 60, 1, 34650, 16800, 2520, 120, 1, 756756, 450450, 92400, 7560, 210, 1, 17153136, 12108096, 3153150, 369600, 18900, 336, 1, 399072960, 325909584, 102918816, 15765750, 1201200, 41580, 504, 1, 9465511770, 8779605120, 3259095840, 617512896, 63063000, 3363360, 83160, 720, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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).

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).

LINKS

Gheorghe Coserea, Rows n=0..100, flattened

FORMULA

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).

G.f. y = Sum_{n>=0} P_n(t)*x^n satisfies:

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.

EXAMPLE

A(x;t) = 1 + (6 + t)*x + (90 + 24*t + t^2)*x^2 + (1680 + 630*t + 60*t^2 + t^3)*x^3 + ...

Triangle starts:

n\k [0]        [1]        [2]        [3]       [4]      [5]    [6]  [7]

[0] 1;

[1] 6,         1;

[2] 90,        24,        1;

[3] 1680,      630,       60,        1;

[4] 34650,     16800,     2520,      120,      1;

[5] 756756,    450450,    92400,     7560,     210,     1;

[6] 17153136,  12108096,  3153150,   369600,   18900,   336,   1;

[7] 399072960, 325909584, 102918816, 15765750, 1201200, 41580, 504, 1;

[8] ...

PROG

(PARI)

T(n, k) = (3*n - 2*k)!/((n-k)!^3*k!);

concat(vector(10, 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) = { \\ apply diffop p (encoded as Pol in Dx) to Ser 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; t='t;

diag(1/(1 - (x+y+z + t*x*y*z)), 11, [x, y, z]) == vector(11, n, P(n-1))

\\ annihilating diffop:

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

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);

0 == apply_diffop(p, y)

*/

CROSSREFS

Cf. A000172, A006480, A081798, A124435, A318108, A318109.

Sequence in context: A009330 A300512 A343622 * A331557 A303675 A266302

Adjacent sequences:  A318104 A318105 A318106 * A318108 A318109 A318110

KEYWORD

nonn,tabl

AUTHOR

Gheorghe Coserea, Sep 18 2018

STATUS

approved

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Last modified October 22 13:15 EDT 2021. Contains 348170 sequences. (Running on oeis4.)