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A137276 Triangle T(n,k), read by rows: T(n,k)= 0 if n-k odd. T(n,k)= 2*(-1)^((n-k)/2)*(2k-n)/(n+k)*binomial((n+k)/2,(n-k)/2) if n-k even. 14
1, 0, 1, 2, 0, 1, 0, 1, 0, 1, -2, 0, 0, 0, 1, 0, -3, 0, -1, 0, 1, 2, 0, -3, 0, -2, 0, 1, 0, 5, 0, -2, 0, -3, 0, 1, -2, 0, 8, 0, 0, 0, -4, 0, 1, 0, -7, 0, 10, 0, 3, 0, -5, 0, 1, 2, 0, -15, 0, 10, 0, 7, 0, -6, 0, 1, 0, 9, 0, -25, 0, 7, 0, 12, 0, -7, 0, 1, -2, 0, 24, 0, -35, 0, 0, 0, 18, 0, -8, 0, 1, 0, -11, 0, 49, 0, -42, 0, -12, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Polynomial coefficients of P(n,x) in increasing powers, read by rows, where P(0,x)=1, P(1,x)=x, P(2,x)=2+x^2, P(3,x)=x+x^3, P(4,x)=-2+x^4, and  P(n,x) = x*P(n-1,x) - P(n-2,x) for n>=5.

The row-reversed version of A135929.

Row sums are repeating 1, 1, 3, 2, -1, -3, -2, 1, 3, 2, -1..., see A138034 and A119910.

LINKS

Table of n, a(n) for n=0..99.

P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31, MR 1439165

FORMULA

T(n,k)= 0 if n-k odd. T(n,k)= 2*(-1)^((n-k)/2)*(2k-n)/(n+k)*binomial((n+k)/2,(n-k)/2) if n-k even.

P(n,x) = x*P(n-1,x)-P(n-2,x), n>=5.

P(n,2*x) = -2*T(n,x)+4*x*U(n-1,x), where T(n,x) is A053120 and U(n,x) is A053117.

EXAMPLE

{1}, = 1

{0, 1}, = x

{2, 0, 1}, = 2+x^2

{0, 1, 0, 1}, = x+x^3

{-2, 0, 0, 0, 1}, = -2+x^4

{0, -3, 0, -1, 0, 1}, = -3x-x^3+x^5

{2, 0, -3, 0, -2, 0, 1},

{0, 5, 0, -2, 0, -3, 0, 1},

{-2, 0, 8, 0, 0, 0, -4, 0, 1},

{0, -7, 0, 10, 0, 3, 0, -5, 0, 1},

{2, 0, -15, 0, 10, 0, 7, 0, -6, 0, 1},

{0, 9, 0, -25, 0, 7, 0, 12, 0, -7, 0, 1}

MAPLE

A137276 := proc(n, k) local nmk, npk; if n = 0 then 1; elif (n-k) mod 2 <> 0 then 0; else nmk := (n-k)/2 ; npk := (n+k)/2 ; (-1)^nmk*(2*k-n)/npk*binomial(npk, nmk) ; fi; end:

seq( seq(A137276(n, k), k=0..n), n=0..13) ;

CROSSREFS

Cf. A123956, A137289.

Sequence in context: A083889 A127523 A116927 * A287234 A140581 A137277

Adjacent sequences:  A137273 A137274 A137275 * A137277 A137278 A137279

KEYWORD

sign,tabl

AUTHOR

Roger L. Bagula and Gary W. Adamson, Mar 13 2008

EXTENSIONS

Fourth row inserted by the Associate Editors of the OEIS, Aug 27 2009

STATUS

approved

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Last modified September 20 15:44 EDT 2017. Contains 292275 sequences.