OFFSET
0,3
COMMENTS
Let v_1,...,v_n be a basis of an n-dimensional vector space V over the field GF(2). Then a(n+1) is the number of subspaces of V that contain the vector v_1 but do not contain v_2,...,v_n. - Geoffrey Critzer, Jul 05 2018
Also number of Stanley graphs on n nodes. For precise definition see Knuth (1997). - Alois P. Heinz, Sep 24 2019
Also the number of naturally labeled posets on [n] with height at most two. - David Bevan, Jul 28 2022; Nov 16 2023
Also the number of sign mappings X:([n] choose 2) -> {+,-} such that for any ordered 3-tuple a<b<c we have X(ab)X(ac)X(bc) not in {+-+,+++}. - Manfred Scheucher, Jan 05 2024
REFERENCES
S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 318.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..115
David Bevan, Gi-Sang Cheon and Sergey Kitaev, On naturally labelled posets and permutations avoiding 12-34, arXiv:2311.08023 [math.CO], 2023.
Lucas Gagnon, The combinatorics of normal subgroups in the unipotent upper triangular group, arXiv:2012.00108 [math.CO], 2020.
D. E. Knuth, Letter to Daniel Ullman and others, Apr 29 1997 [Annotated scanned copy, with permission]
Zvi Rosen and Yan X. Zhang, Convex Neural Codes in Dimension 1, arXiv:1702.06907 [math.CO], 2017. Mentions this sequence.
R. P. Stanley, Problem 10572, The American Mathematical Monthly, 104(2) (1997), 168.
R. P. Stanley and S. C. Locke, Graphs without increasing paths: Solution to Problem 10572, The American Mathematical Monthly, 106(2) (1999), 168.
FORMULA
O.g.f.: A(x) = Sum_{n>=0} x^n / Product_{k=0..n} (1 - (2^k-1)*x).
G.f.: (G(0) - 1)/(x-1) where G(k) = 1 - 1/(1-x*(2^k-1))/(1-x/(x-1/G(k+1))); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 16 2013
a(n) ~ c * 2^(n^2/4), where c = EllipticTheta[3,0,1/2]/QPochhammer[1/2,1/2] = 7.3719688014613... if n is even and c = EllipticTheta[2,0,1/2]/QPochhammer[1/2,1/2] = 7.3719494907662... if n is odd. - Vaclav Kotesovec, Aug 23 2013
a(n) = Sum_{k=0..n} qStirling2(n,k), where qStirling2 is the triangle A139382. - Vladimir Kruchinin, Feb 26 2020
G.f.: f(1), where f(y) = 1 + x*((y-1)*f(y) + f(2*y)). - David Bevan, Jul 28 2022
EXAMPLE
O.g.f.: A(x) = 1 + x/(1-x) + x^2/((1-x)*(1-3x)) + x^3/((1-x)*(1-3x)*(1-7x)) + x^4/((1-x)*(1-3x)*(1-7x)*(1-15x)) + ...
MAPLE
b:= proc(n) option remember; add(mul(
(2^(i+k)-1)/(2^i-1), i=1..n-k), k=0..n)
end:
a:= proc(n) option remember;
add(b(n-j)*binomial(n, j)*(-1)^j, j=0..n)
end:
seq(a(n), n=0..19); # Alois P. Heinz, Sep 24 2019
MATHEMATICA
Table[SeriesCoefficient[Sum[x^n/Product[(1-(2^k-1)*x), {k, 0, n}], {n, 0, nn}], {x, 0, nn}] , {nn, 0, 20}] (* Vaclav Kotesovec, Aug 23 2013 *)
b[n_] := b[n] = Sum[Product[(2^(i+k)-1)/(2^i-1), {i, 1, n-k}], {k, 0, n}];
a[n_] := a[n] = Sum[(-1)^j b[n-j] Binomial[n, j], {j, 0, n}];
a /@ Range[0, 19] (* Jean-François Alcover, Mar 10 2020, after Alois P. Heinz *)
PROG
(PARI) a(n)=polcoeff(sum(k=0, n, x^k/prod(j=0, k, 1-(2^j-1)*x+x*O(x^n))), n)
(Sage) # After Vladimir Kruchinin.
def a(n):
@cached_function
def T(n, k):
if k == 1 or k == n: return 1
return (2^k-1)*T(n-1, k) + T(n-1, k-1)
return sum(T(n, k) for k in (1..n)) if n > 0 else 1
print([a(n) for n in (0..19)]) # Peter Luschny, Feb 26 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 06 2007
EXTENSIONS
References for Stanley graphs added by David Bevan, Jul 24 2024
STATUS
approved