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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| 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.
The (unsigned) column sequences are, for m=0..10: A000774, A081052, A103719-A103727.
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LINKS
| W. Lang, triangular array.
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FORMULA
| 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)-th row of Stirling cycle numbers (unsigned, A008275); say a,b,c,d. Then A, n-th row of present triangle = ((a+b), (b+c), (c+d), (d)). E.g. 4-th 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 (qntmpkt(AT)yahoo.com), May 07 2006
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EXAMPLE
| [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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CROSSREFS
| Cf. A008275.
Sequence in context: A171515 A110271 A073107 * A113350 A164678 A164679
Adjacent sequences: A103715 A103716 A103717 * A103719 A103720 A103721
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KEYWORD
| sign,easy,tabl
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Feb 24 2005
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