A103719 -id:A103719 - cp's OEIS frontend

A103718 Triangle of coefficients of certain polynomials used with prime numbers as variables in the computation of the array A103728.

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

Offset: 0

Wolfdieter Lang, Feb 24 2005

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.

			Triangle begins:
    1;
    2,   -1;
    5,   -4,    1;
   17,  -17,    7,   -1;
   74,  -85,   45,  -11,    1;
  394, -499,  310, -100,   16,   -1;
  ...

W. Lang, Initial section of triangular array.

Cf. A008275.

a[0, 0] = 1; a[k_, 0] := (k - 1)! + k*a[k - 1, 0]; a[k_, m_]:= If[kIndranil Ghosh, Mar 11 2017 *)

a(k, m) = if(m==0, if(k==0, 1, (k - 1)! + k*a(k - 1, 0)) , if(kIndranil Ghosh, Mar 11 2017

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

More terms from Indranil Ghosh, Mar 11 2017

A000454 Unsigned Stirling numbers of first kind s(n,4).

Original entry on oeis.org

1, 10, 85, 735, 6769, 67284, 723680, 8409500, 105258076, 1414014888, 20313753096, 310989260400, 5056995703824, 87077748875904, 1583313975727488, 30321254007719424, 610116075740491776

Offset: 4

N. J. A. Sloane

nonn

Number of permutations of n elements with exactly 4 cycles.

The asymptotic expansion of the higher order exponential integral E(x, m=4, n=1) ~ exp(-x)/x^4*(1 - 10/x + 85/x^2 - 735/x^3 + 6769/x^4 - ...) leads to the sequence given above. See A163931 and A163932 for more information. - Johannes W. Meijer, Jun 11 2016

			(-log(1-x))^4 = x^4 + 2*x^5 + (17/6)*x^6 + (7/2)*x^7 + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 833.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 217.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Shanzhen Gao, Permutations with Restricted Structure (in preparation) [From Shanzhen Gao, Sep 14 2010] [Apparently unpublished as of June 2016]

T. D. Noe, Table of n, a(n) for n=4..100
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 33.

Cf. A000254, A000399, A000482, A001233, A001234, A008275, A243569, A243570.

Abs[StirlingS1[Range[4,20],4]] (* Harvey P. Dale, Aug 26 2011 *)

for(n=3,50,print1(polcoeff(prod(i=1,n,x+i),3,x),","))

[stirling_number1(i,4) for i in range(4,22)] # Zerinvary Lajos, Jun 27 2008

Let P(n-1,X) = (X+1)(X+2)(X+3)...(X+n-1); then a(n) is the coefficient of X^3; or a(n) = P'''(n-1,0)/3!. - Benoit Cloitre, May 09 2002 [Edited by Petros Hadjicostas, Jun 29 2020 to agree with the offset 4]
E.g.f.: (-log(1-x))^4/4!. [Corrected by Joerg Arndt, Oct 05 2009]
a(n) is coefficient of x^(n+4) in (-log(1-x))^4, multiplied by (n+4)!/4!.
a(n) = (h(n-1, 1)^3 - 3*h(n-1, 1)*h(n-1, 2) + 2*h(n-1, 3))*(n-1)!/3!, where h(n, r) = Sum_{i=1..n} 1/i^r. - Klaus Strassburger, 2000
a(n) = det(|S(i+4,j+3)|, 1 <= i,j <= n-4), where S(n,k) are Stirling numbers of the second kind. - Mircea Merca, Apr 06 2013
a(n) = y(n)*n!/24, where y(0) = y(1) = y(2) = y(3) = 0, y(4) = 1 and n^4*y(n) + (-1-5*n-10*n^2-10*n^3-4*n^4)*y(n+1) + (1+n)*(2+n)*(7+12*n+6*n^2)*y(n+2) - 2*(1+n)*(2+n)*(3+n)*(3+2*n)*y(3+n) + (1+n)*(2+n)*(3+n)*(4+n)*y(n+4) = 0. - Benedict W. J. Irwin, Jul 12 2016
From Vaclav Kotesovec, Jul 12 2016: (Start)
a(n) = 2*(2*n - 5)*a(n-1) - (6*n^2 - 36*n + 55)*a(n-2) + (2*n - 7)*(2*n^2 - 14*n + 25)*a(n-3) - (n-4)^4*a(n-4).
a(n) ~ n! * (log(n))^3 / (6*n) * (1 + 3*gamma/log(n) + (3*gamma^2 - Pi^2/2)/ (log(n))^2), where gamma is the Euler-Mascheroni constant A001620. (End)
From Petros Hadjicostas, Jun 29 2020: (Start)
a(n) = A000399(n-1) + (n-1)*a(n-1) for n >= 1 (assuming a(n) = 0 for n = 0..3).
a(n) = A103719(n-4) + (n-2)*a(n-1) for n >= 4.
a(n) = A000254(n-3) + (2*n-3)*a(n-1) - (n-2)^2*a(n-2) for n >= 3.
a(n) = (n-4)! + 3*(n-2)*a(n-1) - (3*n^2-15*n+19)*a(n-2) + (n-3)^3*a(n-3) for n >= 4. (End)

More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 18 2000

cp's OEIS Frontend

A103718 Triangle of coefficients of certain polynomials used with prime numbers as variables in the computation of the array A103728.

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