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

0,1

References

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

Links

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

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)

Example

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;

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 *)

Prog

(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
(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

Crossrefs

Cf. A007283, A028326, A052548.

Sequence in context: A145961 A369557 A082928 * A247413 A108127 A207376

Adjacent sequences: A139521 A139522 A139523 * A139525 A139526 A139527

Keyword

nonn,tabl,easy,less

Author

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

Status

approved