OFFSET
0,5
COMMENTS
After a(3) = 1, always divisible by 3. a(n) is 3 times a prime (A001748) when n = 5, 6, 11, 14, 15, 16, 19. - Jonathan Vos Post, Jun 22 2008
REFERENCES
D. E. Knuth, TAOCP, Vol. 4, Section 7.2.1.7, Problem 26.
George Puttenham, The Arte of English Poesie (1589), page 72, can be said to have stated the problem; but he omitted one case for n=5 and 22 cases for n=6, so he must have had other constraints in mind!
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..500
FORMULA
If f(z) is the generating function for A074664, then a(z)=zf(z)/(1+z-f(z)).
Also, if g(z) is the generating function for A000296, then a(z) = 1-1/g(z).
O.g.f.: x^2/(1-x-2*x^2/(1-2*x-3*x^2/(1-3*x-4*x^2/(1-... -n*x-(n+1)*x^2/(1- ...)))))) (continued fraction). - Paul D. Hanna, Jan 17 2006
From Sergei N. Gladkovskii, Sep 20 2012, Nov 04 2012, Feb 04 2013, Feb 23 2013, Apr 18 2013, May 12 2013: (Start) Continued fractions:
G.f.: -x + 2*x/E(0) where E(k)= 1 + 1/(1 + 2*x/(1 - 2*(k+2)*x/E(k+1))).
G.f.: 1 - x*U(0,1/x) where U(k,x)= x - k - (k+1)/U(k+1,x).
G.f.: (1+x)*x/G(0) - x where G(k) = 1 + x - x*(k+1)/(1 - x/G(k+1)).
G.f.: x/Q(0) - x where Q(k)= 1 + x/(x*k-x-1)/Q(k+1).
G.f.: 1 - Q(0) where Q(k)= 1 + x - x/(1 - x*(k+1)/Q(k+1)).
G.f.: 1-x-1/Q(0) where Q(k)= 1 + x/(1 - x - x*(k+1)/(x + 1/Q(k+1))). (End)
EXAMPLE
a(5)=9 because of the set partitions 135|24, 134|25, 125|34, 145|23, 15|234, 13|245, 124|35, 12345, 14|235. [Puttenham missed the last of these.]
MAPLE
F:= proc(n) option remember; convert(series(1 -1/add(coeff(series(exp(exp(x)-1), x, n+1), x, j)*j!*x^j, j=0..n), x, n+1), polynom) end: a:= n-> coeff(series(x*F(n)/(1+x-F(n)), x, n+1), x, n): seq(a(n), n=0..24); # Alois P. Heinz, Sep 05 2008
MATHEMATICA
f[n_] := f[n] = Normal[ Series[ 1-1/Sum[ SeriesCoefficient[ Series[ Exp[Exp[x] - 1], {x, 0, n + 1}], {x, 0, j}]*j!*x^j, {j, 0, n}], {x, 0, n + 1}]]; a[0] = 0; a[n_] := SeriesCoefficient[ Series[ x*(f[n]/(1 + x - f[n])), {x, 0, n + 1}], {x, 0, n}]; Table[a[n], {n, 0, 24}] (* Jean-François Alcover, translated from Maple *)
PROG
(Sage)
def A098742_list(dim):
T = matrix(ZZ, dim, dim)
for n in range(dim): T[n, n] = 1
for n in (1..dim-1):
for k in (0..n-1):
T[n, k] = T[n-1, k-1]+(k+1)*T[n-1, k]+(k+2)*T[n-1, k+1]
return [0, 0]+list(T.column(0))
A098742_list(23) # - Peter Luschny, Sep 20 2012
CROSSREFS
KEYWORD
nice,nonn,easy
AUTHOR
Don Knuth, Oct 01 2004
EXTENSIONS
More terms from Vladeta Jovovic, Oct 21 2004
STATUS
approved