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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

Table of n, a(n) for n=0..54.

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<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 *)

PROG

(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

CROSSREFS

Cf. A008275.

Sequence in context: A110271 A073107 A248669 * A113350 A227372 A164678

Adjacent sequences:  A103715 A103716 A103717 * A103719 A103720 A103721

KEYWORD

sign,easy,tabl

AUTHOR

Wolfdieter Lang, Feb 24 2005

EXTENSIONS

More terms from Indranil Ghosh, Mar 11 2017

STATUS

approved

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Last modified November 19 11:04 EST 2017. Contains 294936 sequences.