This site is supported by donations to The OEIS Foundation.

 Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS". Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A103718 Triangle of coefficients of certain polynomials used with prime numbers as variables in the computation of the array A103728. 21
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 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. LINKS W. Lang, triangular array. 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., 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 EXAMPLE Triangle begins:     1;     2,   -1;     5,   -4,    1;    17,  -17,    7,   -1;    74,  -85,   45,  -11,    1;   394, -499,  310, -100,   16,   -1;   ... MATHEMATICA a[0, 0] = 1; a[k_, 0] := (k - 1)! + k*a[k - 1, 0]; a[k_, m_]:= If[k

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.