A172497 - OEIS

%I #10 Apr 21 2021 17:29:16

%S 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,

%T 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,

%U 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

%N 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.

%H G. C. Greubel, <a href="/A172497/b172497.txt">Rows n = 0..50 of the triangle, flattened</a>

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

%e The triangle begins as:

%e   1;

%e   1, 1;

%e   1, 1,  1;

%e   1, 1,  1,  1;

%e   1, 2,  2,  2,  1;

%e   1, 1,  2,  2,  1,  1;

%e   1, 2,  2,  4,  2,  2,  1;

%e   1, 3,  6,  6,  6,  6,  3,  1;

%e   1, 2,  6, 12,  6, 12,  6,  2,  1;

%e   1, 4,  8, 24, 24, 24, 24,  8,  4, 1;

%e   1, 3, 12, 24, 36, 72, 36, 24, 12, 3, 1;

%t 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];

%t c[n_]:= Product[b[[j]], {j,n}];

%t T[n_, k_]:= Round[c[n]/(c[k]*c[n-k])];

%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Apr 20 2021 *)

%o (Magma)

%o R<x>:= PowerSeriesRing(Integers(), 100);

%o b:= Coefficients(R!( 1/(1+x-x^3-x^4-x^5-x^6-x^7+x^9+x^10) ));

%o c:= func< n | (&*[b[j]: j in [10..n+10]]) >;

%o T:= func< n,k | Round(c(n)/(c(k)*c(n-k))) >;

%o [T(n,k): k in [0..n], n in [1..12]]; // _G. C. Greubel_, Apr 20 2021

%o (Sage)

%o @CachedFunction

%o def A029826_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     return P( 1/(1+x-x^3-x^4-x^5-x^6-x^7+x^9+x^10) ).list()

%o b=A029826_list(130)

%o def c(n): return product(b[j] for j in (9..n+9))

%o def T(n,k): return round(c(n)/(c(k)*c(n-k)))

%o flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Apr 20 2021

%Y Cf. A029826.

%K nonn,tabl,easy,less

%O 0,12

%A _Roger L. Bagula_, Feb 05 2010

%E Definition corrected and edited by _G. C. Greubel_, Apr 20 2021