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A103718
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Triangle of coefficients of certain polynomials used with prime numbers as variables in the computation of the array A103728.
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21
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1, 2, -1, 5, -4, 1, 17, -17, 7, -1, 74, -85, 45, -11, 1, 394, -499, 310, -100, 16, -1, 2484, -3388, 2359, -910, 196, -22, 1, 18108, -26200, 19901, -8729, 2282, -350, 29, -1, 149904, -227708, 185408, -89733, 26985, -5082, 582, -37, 1, 1389456, -2199276, 1896380, -993005, 332598, -72723, 10320, -915, 46, -1
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OFFSET
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0,2
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COMMENTS
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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).
Row sums (signed) give A000142(k)=k!. Row sums (unsigned) coincide with A007680(k)=(2*k+1)*k!, k>=0.
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LINKS
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FORMULA
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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.
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.
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
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EXAMPLE
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Triangle begins:
1;
2, -1;
5, -4, 1;
17, -17, 7, -1;
74, -85, 45, -11, 1;
394, -499, 310, -100, 16, -1;
...
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MATHEMATICA
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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 *)
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PROG
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(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)));
{for(k=0, 9, for(m=0, k, print1(a(k, m), ", "); ); print(); ); } \\ Indranil Ghosh, Mar 11 2017
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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