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A181543
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Triangle of cubed binomial coefficients, T(n,k) = C(n,k)^3, read by rows.
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16
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1, 1, 1, 1, 8, 1, 1, 27, 27, 1, 1, 64, 216, 64, 1, 1, 125, 1000, 1000, 125, 1, 1, 216, 3375, 8000, 3375, 216, 1, 1, 343, 9261, 42875, 42875, 9261, 343, 1, 1, 512, 21952, 175616, 343000, 175616, 21952, 512, 1, 1, 729, 46656, 592704, 2000376, 2000376, 592704, 46656, 729, 1
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OFFSET
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0,5
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COMMENTS
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Diagonal of rational function R(x,y,z,t) = 1/(1 + y + z + x*y + y*z + t*x*z + (t+1)*x*y*z) with respect to x, y, z, i.e., T(n,k) = [(xyz)^n*t^k] R(x,y,z,t). - Gheorghe Coserea, Jul 01 2018
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LINKS
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FORMULA
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Row sums equal A000172, the Franel numbers.
Central terms are A002897(n) = C(2n,n)^3.
The g.f. of the antidiagonal sums is Sum_{n>=0} (3n)!/(n!)^3 * x^(3n)/(1-x-x^2)^(3n+1).
G.f. for column k: [Sum_{j=0..2k} A181544(k,j)*x^j]/(1-x)^(3k+1), where the row sums of A181544 equals De Bruijn's s(3,n) = (3n)!/(n!)^3.
G.f.: A(x,y) = Sum_{n>=0} (3n)!/n!^3 * x^(2n)*y^n/(1-x-x*y)^(3n+1). - Paul D. Hanna, Nov 04 2010
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EXAMPLE
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Triangle begins:
1;
1, 1;
1, 8, 1;
1, 27, 27, 1;
1, 64, 216, 64, 1;
1, 125, 1000, 1000, 125, 1;
1, 216, 3375, 8000, 3375, 216, 1;
1, 343, 9261, 42875, 42875, 9261, 343, 1;
1, 512, 21952, 175616, 343000, 175616, 21952, 512, 1;
1, 729, 46656, 592704, 2000376, 2000376, 592704, 46656, 729, 1;
...
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MAPLE
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T:= (n, k)-> binomial(n, k)^3:
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MATHEMATICA
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Flatten[Table[Binomial[n, k]^3, {n, 0, 10}, {k, 0, n}]] (* Harvey P. Dale, May 23 2011 *)
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PROG
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(PARI) T(n, k)=binomial(n, k)^3
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print())
(PARI) T(n, k)=polcoeff(polcoeff(sum(m=0, n, (3*m)!/m!^3*x^(2*m)*y^m/(1-x-x*y+x*O(x^n))^(3*m+1)), n, x), k, y)
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print()) \\ Paul D. Hanna, Nov 04 2010
(PARI)
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] = polcoeff(a[n], n-1)));
return(a);
};
x='x; y='y; z='z; t='t;
concat(apply(Vec, diag(1/(1 + y + z + x*y + y*z + t*x*z + (t+1)*x*y*z), 10, [x, y, z]))) \\ Gheorghe Coserea, Jul 01 2018
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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