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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.
1
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
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
KEYWORD
nonn,tabl,easy,less
AUTHOR
STATUS
approved