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%I #27 Aug 28 2019 15:53:09
%S 1,2,-1,5,-4,1,17,-17,7,-1,74,-85,45,-11,1,394,-499,310,-100,16,-1,
%T 2484,-3388,2359,-910,196,-22,1,18108,-26200,19901,-8729,2282,-350,29,
%U -1,149904,-227708,185408,-89733,26985,-5082,582,-37,1,1389456,-2199276,1896380,-993005,332598,-72723,10320,-915,46,-1
%N Triangle of coefficients of certain polynomials used with prime numbers as variables in the computation of the array A103728.
%C The g.f. for the sequence {b(N,p)}, with b(N,p) the number of cyclically inequivalent two-color, N bead necklaces with p beads of one color and N-p beads of the other color is, for prime numbers p, G(p(n),x):=P(p(n)-1,x)/((1-x)^(p(n)-1)*(1-x^p(n))), with the numerator polynomial P(p(n)-1,x):= sum(r(n,k)*x^k,k=0..p(n)-1) and the row polynomials of this triangle r(n,k):=sum(a(k,m)*p(n)^m,m=0..k). p(n)=A000040(n) (prime numbers).
%C Row sums (signed) give A000142(k)=k!. Row sums (unsigned) coincide with A007680(k)=(2*k+1)*k!, k>=0.
%C The (unsigned) column sequences are, for m=0..10: A000774, A081052, A103719-A103727.
%H W. Lang, <a href="/A103718/a103718.txt">Initial section of triangular array.</a>
%F a(k, m) = ((-1)^m)*(|S1(k+1, m+1)| + |S1(k+1, m+2)|) = ((-1)^m)*(|S1(k+2, m+2)|-k*|S1(k+1, m+2)|), with the (signed) Stirling number triangle S1(n, m) = A048994(n, m), n >= m >= 0.
%F a(0, 0)=1, a(k, 0) = (k-1)! + k*a(k-1, 0); a(k, m) = -a(k-1, m-1) + k*a(k-1, m), m > 0 and a(k, m)=0 if k < m.
%F Let B = (n+1)-st row of Stirling cycle numbers (unsigned, A008275); say a,b,c,d,.... Then n-th row of present triangle = ((a+b), (b+c), (c+d), ..., (d)). E.g., 4th row of the Stirling cycle numbers = (6, 11, 6, 1). Then third row of A103718 = ((6+11), (11+6), (6+1), (1)) = (17, 17, 7, 1). - _Gary W. Adamson_, May 07 2006
%e Triangle begins:
%e 1;
%e 2, -1;
%e 5, -4, 1;
%e 17, -17, 7, -1;
%e 74, -85, 45, -11, 1;
%e 394, -499, 310, -100, 16, -1;
%e ...
%t a[0, 0] = 1; a[k_, 0] := (k - 1)! + k*a[k - 1, 0]; a[k_, m_]:= If[k<m, 0, -a[k - 1, m - 1] + k a[k - 1, m]]; Flatten[Table[a[k, m], {k, 0, 9}, {m, 0, k}]] (* _Indranil Ghosh_, Mar 11 2017 *)
%o (PARI) a(k, m) = if(m==0, if(k==0, 1, (k - 1)! + k*a(k - 1, 0)) , if(k<m, 0, -a(k - 1, m - 1) + k*a(k - 1, m)));
%o {for(k=0, 9, for(m=0, k, print1(a(k, m),", ");); print(););} \\ _Indranil Ghosh_, Mar 11 2017
%Y Cf. A008275.
%K sign,easy,tabl
%O 0,2
%A _Wolfdieter Lang_, Feb 24 2005
%E More terms from _Indranil Ghosh_, Mar 11 2017
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