A135922 Inverse binomial transform of A006116, which is the sum of Gaussian binomial coefficients [n,k] for q=2.

1, 1, 2, 6, 26, 158, 1330, 15414, 245578, 5382862, 162700898, 6801417318, 394502066810, 31849226170622, 3589334331706258, 566102993389615254, 125225331231990004138, 38920655753545108286254, 17021548688670112527781058, 10486973328106497739526535366

Offset: 0

Author

Paul D. Hanna, Dec 06 2007

Keywords

• nonn

Comments

Let B = {v_1,v_2,...,v_n} be a basis for F_2^n. a(n) is the number of subspaces of F_2^n that do not contain any of the vectors in B. Moreover, for 1<=k<=n, let S be a size k subset of B. a(k) is the number of subspaces of F_2^n that do not contain any of the vectors in S but do contain all the vectors in B - S. - Geoffrey Critzer, May 03 2025

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 aManfred Scheucher, Jan 05 2024

Examples

			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)) + ...
		

References

• Steven 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.

Crossrefs

Cf. A006116, A323841, A323842, A323843, A139382, A006455, A356111.

Programs

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 *)

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

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))); (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

E.g.f.: exp(-x)*g(x) where g(x) is the e.g.f. for A006116. (given in D. E. Knuth link) - Geoffrey Critzer, May 03 2025

Extensions

References for Stanley graphs added by David Bevan, Jul 24 2024