A172497 Triangle T(n, k) = round( c(n)/(c(k)*c(n-k)) ) where c(n) = Product_{j=1..n} A029826(j+10), read by rows.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 4, 2, 2, 1, 1, 3, 6, 6, 6, 6, 3, 1, 1, 2, 6, 12, 6, 12, 6, 2, 1, 1, 4, 8, 24, 24, 24, 24, 8, 4, 1, 1, 3, 12, 24, 36, 72, 36, 24, 12, 3, 1, 1, 5, 15, 60, 60, 180, 180, 60, 60, 15, 5, 1, 1, 5, 25, 75, 150, 300, 450, 300, 150, 75, 25, 5, 1

Offset: 0

Author

Roger L. Bagula, Feb 05 2010

Keywords

• nonn
• tabl
• easy
• less

Examples

			The triangle begins as:
  1;
  1, 1;
  1, 1,  1;
  1, 1,  1,  1;
  1, 2,  2,  2,  1;
  1, 1,  2,  2,  1,  1;
  1, 2,  2,  4,  2,  2,  1;
  1, 3,  6,  6,  6,  6,  3,  1;
  1, 2,  6, 12,  6, 12,  6,  2,  1;
  1, 4,  8, 24, 24, 24, 24,  8,  4, 1;
  1, 3, 12, 24, 36, 72, 36, 24, 12, 3, 1;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A029826.

Programs

Magma

R:= PowerSeriesRing(Integers(), 100);
b:= Coefficients(R!( 1/(1+x-x^3-x^4-x^5-x^6-x^7+x^9+x^10) ));
c:= func< n | (&*[b[j]: j in [10..n+10]]) >;
T:= func< n,k | Round(c(n)/(c(k)*c(n-k))) >;
[T(n,k): k in [0..n], n in [1..12]]; // G. C. Greubel, Apr 20 2021

Mathematica

b:= Drop[CoefficientList[Series[1/(1+x-x^3-x^4-x^5-x^6-x^7+x^9+x^10), {x,0,100}], x], 10];
c[n_]:= Product[b[[j]], {j,n}];
T[n_, k_]:= Round[c[n]/(c[k]*c[n-k])];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Apr 20 2021 *)

Sage

@CachedFunction
def A029826_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1+x-x^3-x^4-x^5-x^6-x^7+x^9+x^10) ).list()
b=A029826_list(130)
def c(n): return product(b[j] for j in (9..n+9))
def T(n,k): return round(c(n)/(c(k)*c(n-k)))
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 20 2021

Formula

T(n, k) = round( c(n)/(c(k)*c(n-k)) ) where c(n) = Product_{j=1..n} A029826(j+10).

Extensions

Definition corrected and edited by G. C. Greubel, Apr 20 2021