A139524 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.

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, 2, 130, 14, 42, 70, 70, 42, 14, 2, 258, 16, 56, 112, 140, 112, 56, 16, 2, 514, 18, 72, 168, 252, 252, 168, 72, 18, 2, 1026, 20, 90, 240, 420, 504, 420, 240, 90, 20, 2

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Jun 09 2008

Keywords

• nonn
• tabl
• easy
• less

Examples

			Triangle begins as:
     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,   2;
   130, 14, 42,  70,  70,  42,  14,   2;
   258, 16, 56, 112, 140, 112,  56,  16,  2;
   514, 18, 72, 168, 252, 252, 168,  72, 18,  2;
  1026, 20, 90, 240, 420, 504, 420, 240, 90, 20, 2;
		

References

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

Links

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

Crossrefs

Cf. A007283, A028326, A052548.

Programs

Magma

A139524:= func< n,k | k eq 0 select 2+2^n else 2*Binomial(n,k) >;
[A139524(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 02 2021

Mathematica

(* First program *)
T[n_, k_]:= SeriesCoefficient[Series[2*(1+x)^n + 2^n, {x, 0, 20}], k];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, May 02 2021 *)
(* Second program *)
T[n_, k_]:= T[n, k] = If[k==0, 2 + 2^n, 2*Binomial[n, k]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 02 2021 *)

Sage

def A139524(n,k): return 2+2^n if (k==0) else 2*binomial(n,k)
flatten([[A139524(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 02 2021

Formula

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

From R. J. Mathar, Sep 12 2013: (Start)

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

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