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 A181543 Triangle of cubed binomial coefficients, T(n,k) = C(n,k)^3, read by rows. 13
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS 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 LINKS Indranil Ghosh, Rows 0..120 of triangle, flattened Jeffrey S. Geronimo, Hugo J. Woerdeman, and Chung Y. Wong, The autoregressive filter problem for multivariable degree one symmetric polynomials, arXiv:2101.00525 [math.CA], 2021. FORMULA Row sums equal A000172, the Franel numbers. Central terms are A002897(n) = C(2n,n)^3. Antidiagonal sums equal A181545; 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 EXAMPLE 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;   ... MAPLE T:= (n, k)-> binomial(n, k)^3: seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Jan 06 2021 MATHEMATICA Flatten[Table[Binomial[n, k]^3, {n, 0, 10}, {k, 0, n}]] (* Harvey P. Dale, May 23 2011 *) PROG (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 CROSSREFS Cf. A000172 (row sums), A181545 (antidiagonal sums), A002897, A181544, A248658. Variants: A008459, A007318. Sequence in context: A220718 A176283 A323324 * A141696 A178122 A142470 Adjacent sequences:  A181540 A181541 A181542 * A181544 A181545 A181546 KEYWORD nonn,tabl AUTHOR Paul D. Hanna, Oct 30 2010 STATUS approved

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Last modified April 20 23:46 EDT 2021. Contains 343143 sequences. (Running on oeis4.)