OFFSET
6,1
COMMENTS
Associated Stirling numbers.
From Enrique Navarrete, May 24 2025: (Start)
6*a(n) is the number of ternary words of length n that contain at least two of each of the symbols of the alphabet. For example, 6*a(6) counts the 90 permutations of 001122.
2*a(n+1) is the number of ternary strings of length n that contain at least one 0 and at least two 1's and at least two 2's. For example, for n = 6, 2*a(7) counts the 90 permutations of 001122, the 60 permutations of 011122, and the 60 permutations of 011222. (End)
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 222.
F. N. David and D. E. Barton, Combinatorial Chance. Hafner, NY, 1962, p. 296.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 76.
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).
LINKS
T. D. Noe, Table of n, a(n) for n=6..200
Daniel J. Bernstein, Andreas Hülsing, Decisional second-preimage resistance: When does SPR imply PRE?, (2019).
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Erik Vigren and Andreas Dieckmann, A New Result in Form of Finite Triple Sums for a Series from Ramanujan's Notebooks, Symmetry (2022) Vol. 14, No. 6, 1090.
Index entries for linear recurrences with constant coefficients, signature (10,-40,82,-91,52,-12).
FORMULA
E.g.f.: ((exp(x) - 1 - x)^3)/3!.
G.f.: x^6*(12*x^3 - 40*x^2 + 45*x - 15)/((1 - x)^3*(1 - 2*x)^2*(3*x - 1)). - Simon Plouffe in his 1992 dissertation
a(n) = (1+n+n^2)/2 - (1/2 + n/4)*2^n + 3^n/6. - Michael Steyer (m.steyer(AT)osram.de), Jan 09 2005
a(n) = 10*a(n-1) - 40*a(n-2) + 82*a(n-3) - 91*a(n-4) + 52*a(n-5) - 12*a(n-6), n > 11. - Harvey P. Dale based on Michael Steyer's formula, Jul 17 2011
a(n) = 3*a(n-1) + (2^(n-3)-n+1)*(n-1), a(n)=0, n < 6. - Vladimir Kruchinin, Oct 04 2018
EXAMPLE
a(6) = 6!/(2!*2!*2!*3!) = 15.
MATHEMATICA
Table[(1+n+n^2)/2-(1/2+n/4)*2^n+3^n/6, {n, 6, 30}] (* or *) LinearRecurrence[ {10, -40, 82, -91, 52, -12}, {15, 105, 490, 1918, 6825, 22935}, 25] (* Harvey P. Dale, Jul 17 2011 *)
offset = 6; terms = 26; egf = (Exp[x]-1-x)^3/3!; Drop[CoefficientList[egf + O[x]^(terms+offset), x]*Range[0, terms+offset-1]!, offset] (* Jean-François Alcover, May 07 2017 *)
PROG
(PARI) Vec(x^6*(12*x^3-40*x^2+45*x-15)/((1-x)^3*(1-2*x)^2*(3*x-1))+O(x^66)) /* Joerg Arndt, Apr 10 2013 */
(Python) # based on Vladimir Kruchinin's formula
def A000478():
a = 15; n = 7; z = 4; s = 15;
while True:
yield a
z = 2*z; s += n*(z-2) + 3; a = 3*a + s; n += 1
a = A000478(); print([next(a) for _ in range(6, 32)]) # Peter Luschny, Oct 04 2018
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
Additional comments from Michael Steyer, Dec 02 2000
More terms from James Sellers, Dec 06 2000
More terms from Joerg Arndt, Apr 10 2013
STATUS
approved