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A201461 Triangle read by rows: n-th row (n>=0) gives coefficients of the polynomial ((x+1)^(2^n) + (x-1)^(2^n))/2. 4
1, 1, 1, 1, 6, 1, 1, 28, 70, 28, 1, 1, 120, 1820, 8008, 12870, 8008, 1820, 120, 1, 1, 496, 35960, 906192, 10518300, 64512240, 225792840, 471435600, 601080390, 471435600, 225792840, 64512240, 10518300, 906192, 35960, 496, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Wanted: reference for the fact that these polynomials are irreducible. Washington, Cyclotomic Fields, perhaps?

The algorithm r(n) = (1/2)*(r(n-1) + A/r(n-1)), starting with r(0) = A, used for approximating sqrt(A), which is known as the Babylonian method or Hero's method after the first-century Greek mathematician Hero of Alexandria and which can be derived from Newton's method, generates fractions beginning with (A+1)/2, (A^2 + 6*A + 1)/(4*(A+1)), (A^4 + 28*A^3 + 70*A^2 + 28*A + 1)/(8*(A+1)*(A^2 + 6*A + 1)), ... This is p(n,sqrt(A))/(2^n*Product_{k=1..n-1} p(k,sqrt(A))) with the given polynomial p(n,x) = ((x+1)^(2^n) + (x-1)^(2^n))/2. - Martin Renner, Jan 11 2017

The quadratic coefficient of this polynomial is A006516(n), the even-indexed coefficients are binomial(2^n,2*k) or A086645(2^(n-1),k) for 0 <= k <= 2^(n-1), in each row the maximum central coefficient for n>=2 is A037293(n) or A000984(2^(n-1)). - Martin Renner, Jan 14 2017

T(n,k) and A281122 are a bisection of row 2^n of Pascal's triangle A007318. - Martin Renner, Jan 15 2017

LINKS

Indranil Ghosh, Rows 0..11, flattened

FORMULA

T(n,k) = binomial(2^n,2*k). - Joerg Arndt, Jan 15 2017

EXAMPLE

The first few polynomials are

1,

x^2 + 1,

x^4 + 6*x^2 + 1,

x^8 + 28*x^6 + 70*x^4 + 28*x^2 + 1,

x^16 + 120*x^14 + 1820*x^12 + 8008*x^10 + 12870*x^8 + 8008*x^6 + 1820*x^4 + 120*x^2 + 1,

...

The triangle of coefficients begins

1

1, 1

1, 6, 1

1, 28, 70, 28, 1

1, 120, 1820, 8008, 12870, 8008, 1820, 120, 1

1, 496, 35960, 906192, 10518300, 64512240, 225792840, 471435600, 601080390, 471435600, 225792840, 64512240, 10518300, 906192, 35960, 496, 1

...

MATHEMATICA

Flatten[Table[Binomial[2^n, 2k], {n, 0, 6}, {k, 0, 2^(n-1)}]] (* Indranil Ghosh, Feb 22 2017 *)

PROG

(PARI) row(n) = my(v = Vec(((x+1)^(2^n)+(x-1)^(2^n))/2)); vector(#v\2 + 1, k, v[2*k-1]); \\ Michel Marcus, Jan 14 2017

(PARI) T(n, k)=binomial(2^n, 2*k);

for(n=0, 5, for(k=0, 2^(n-1), print1(T(n, k), ", ")); print()); \\ Joerg Arndt, Jan 15 2017

CROSSREFS

Cf. A000984, A006516, A007318, A037293, A086645, A281122.

Sequence in context: A155908 A105373 A296548 * A265603 A174186 A111578

Adjacent sequences:  A201458 A201459 A201460 * A201462 A201463 A201464

KEYWORD

nonn,easy,tabf

AUTHOR

N. J. A. Sloane, Dec 01 2011

STATUS

approved

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Last modified March 26 04:32 EDT 2019. Contains 321481 sequences. (Running on oeis4.)