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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]).

%I A122610 #22 Apr 12 2020 09:33:39
%S A122610 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,
%T A122610 1,1,-8,21,-15,-15,18,1,-2,1,-9,28,-29,-15,39,-6,-9,1,1,-10,36,-49,-7,
%U A122610 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
%N 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]).
%H A122610 P. Steinbach, <a href="http://www.jstor.org/stable/2691048">Golden fields: a case for the heptagon</a>, Math. Mag. 70 (1997), no. 1, 22-31.
%e A122610 1;
%e A122610 1, -2;
%e A122610 1, -3,  1;
%e A122610 1, -4,  3,   1;
%e A122610 1, -5,  6,   1,  -2;
%e A122610 1, -6, 10,  -1,  -6,  1;
%e A122610 1, -7, 15,  -6, -11,  6, 1;
%e A122610 1, -8, 21, -15, -15, 18, 1, -2;
%t A122610 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]
%o A122610 (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))}
%o A122610 (Sage)
%o A122610 @CachedFunction
%o A122610 def T(n,k):
%o A122610     if n< 0: return 0
%o A122610     if n==0: return 1 if k == 0 else 0
%o A122610     h = 2*T(n-1,k) if n==1 else T(n-1,k)
%o A122610     return T(n-1,k-1) - T(n-2,k) - h
%o A122610 A122610 = lambda n,k: T(n,n-k)
%o A122610 for n in (0..9): [A122610(n,k) for k in (0..n)] # _Peter Luschny_, Nov 20 2012
%Y A122610 Cf. A066170.
%K A122610 sign,tabl
%O A122610 0,3
%A A122610 _Roger L. Bagula_ and _Gary W. Adamson_, Sep 20 2006
%E A122610 Edited by _N. J. A. Sloane_, Sep 24 2006
%E A122610 Offset set to 0 by _Michel Marcus_, Feb 07 2014