login
Triangle T(n,k) read by rows: the coefficient of [x^k] of the polynomial 2*(x+1)^n + 2^n in row n, column k.
1

%I #10 May 03 2021 01:26:02

%S 3,4,2,6,4,2,10,6,6,2,18,8,12,8,2,34,10,20,20,10,2,66,12,30,40,30,12,

%T 2,130,14,42,70,70,42,14,2,258,16,56,112,140,112,56,16,2,514,18,72,

%U 168,252,252,168,72,18,2,1026,20,90,240,420,504,420,240,90,20,2

%N Triangle T(n,k) read by rows: the coefficient of [x^k] of the polynomial 2*(x+1)^n + 2^n in row n, column k.

%D Advanced Number Theory, Harvey Cohn, Dover Books, 1963, Pages 88-89

%H G. C. Greubel, <a href="/A139524/b139524.txt">Rows n = 0..50 of the triangle, flattened</a>

%F Sum_{k=0..n} T(n,k) = 3*2^n = A007283(n).

%F From _R. J. Mathar_, Sep 12 2013: (Start)

%F T(n,0) = 2 + 2^n = A052548(n).

%F T(n,k) = 2*binomial(n,k) = A028326(n,k) if k>0. (End)

%e Triangle begins as:

%e 3;

%e 4, 2;

%e 6, 4, 2;

%e 10, 6, 6, 2;

%e 18, 8, 12, 8, 2;

%e 34, 10, 20, 20, 10, 2;

%e 66, 12, 30, 40, 30, 12, 2;

%e 130, 14, 42, 70, 70, 42, 14, 2;

%e 258, 16, 56, 112, 140, 112, 56, 16, 2;

%e 514, 18, 72, 168, 252, 252, 168, 72, 18, 2;

%e 1026, 20, 90, 240, 420, 504, 420, 240, 90, 20, 2;

%t (* First program *)

%t T[n_, k_]:= SeriesCoefficient[Series[2*(1+x)^n + 2^n, {x, 0, 20}], k];

%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, May 02 2021 *)

%t (* Second program *)

%t T[n_, k_]:= T[n, k] = If[k==0, 2 + 2^n, 2*Binomial[n, k]];

%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, May 02 2021 *)

%o (Magma)

%o A139524:= func< n,k | k eq 0 select 2+2^n else 2*Binomial(n,k) >;

%o [A139524(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, May 02 2021

%o (Sage)

%o def A139524(n,k): return 2+2^n if (k==0) else 2*binomial(n,k)

%o flatten([[A139524(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, May 02 2021

%Y Cf. A007283, A028326, A052548.

%K nonn,tabl,easy,less

%O 0,1

%A _Roger L. Bagula_ and _Gary W. Adamson_, Jun 09 2008