A122610 - OEIS

A122610

Triangle read by rows: T(n,k) is coefficient of x^k in Sum_{m=0..n} x^m*(1-x)^(n-m)*(-1)^[(m+1)/2]*binomial(m-[(m+1)/2],[m/2]).

1, 1, -2, 1, -3, 1, 1, -4, 3, 1, 1, -5, 6, 1, -2, 1, -6, 10, -1, -6, 1, 1, -7, 15, -6, -11, 6, 1, 1, -8, 21, -15, -15, 18, 1, -2, 1, -9, 28, -29, -15, 39, -6, -9, 1, 1, -10, 36, -49, -7, 69, -30, -21, 9, 1, 1, -11, 45, -76, 14, 105, -84, -30, 36, 1, -2, 1, -12, 55, -111, 54, 140, -182, -15, 96, -14, -12, 1, 1, -13, 66, -155

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OFFSET

0,3

LINKS

Table of n, a(n) for n=0..81.

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

EXAMPLE

1, -2;

1, -3, 1;

1, -4, 3, 1;

1, -5, 6, 1, -2;

1, -6, 10, -1, -6, 1;

1, -7, 15, -6, -11, 6, 1;

1, -8, 21, -15, -15, 18, 1, -2;

MATHEMATICA

T[n_, k_] := (-1)^Floor[(k + 1)/2]*Binomial[n - Floor[(k + 1)/2], Floor[k/2]]; a = Table[CoefficientList[Sum[T[n, k]*p^k*(1 - p)^(n -k), {k, 0, n}], p], {n, 0, 10}]; Flatten[a]

PROG

(PARI) {T(n, k)=local(A); if(k<0||k>n, 0, A=sum(k=0, n, x^k*(1-x)^(n-k)*(-1)^((k+1)\2)*binomial(n-((k+1)\2), k\2)); polcoeff(A, k))}

(Sage)

@CachedFunction

def T(n, k):

if n< 0: return 0

if n==0: return 1 if k == 0 else 0

h = 2*T(n-1, k) if n==1 else T(n-1, k)

return T(n-1, k-1) - T(n-2, k) - h

A122610 = lambda n, k: T(n, n-k)

for n in (0..9): [A122610(n, k) for k in (0..n)] # Peter Luschny, Nov 20 2012

CROSSREFS