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

0,12

Links

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

Formula

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

Example

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;

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 *)

Prog

(Magma)
R<x>:= 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
(Sage)
@CachedFunction
def A029826_list(prec):
    P.<x> = 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

Crossrefs

Cf. A029826.

Sequence in context: A238408 A048858 A246465 * A211226 A306366 A135265

Adjacent sequences: A172494 A172495 A172496 * A172498 A172499 A172500

Keyword

nonn,tabl,easy,less

Author

Roger L. Bagula, Feb 05 2010

Extensions

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

Status

approved