cp's OEIS Frontend

A007047 Number of chains in power set of n-set.

%I A007047 M2903 #153 Jul 02 2025 10:54:58
%S A007047 1,3,11,51,299,2163,18731,189171,2183339,28349043,408990251,
%T A007047 6490530291,112366270379,2107433393523,42565371881771,921132763911411,
%U A007047 21262618727925419,521483068116543603,13542138653027381291,371206349277313644531
%N A007047 Number of chains in power set of n-set.
%C A007047 Stirling transform of A052849(n-1) = [1,2,4,12,48,...] is a(n-1) =[1,3,11,51,299,...]. - _Michael Somos_, Mar 04 2004
%C A007047 It is interesting to note that a chain in the power set of a set X can be thought of as a fuzzy subset of X and conversely. Chains originating with empty set are fuzzy subsets with empty core and those chains not ending with the whole set are with support strictly contained in X. - Venkat Murali (v.murali(AT)ru.ac.za), May 18 2005
%C A007047 Equals the binomial transform of A000629: (1, 2, 6, 26, 150, 1082, ...) and the double binomial transform of A000670: (1, 1, 3, 13, 75, 541, ...). - _Gary W. Adamson_, Aug 04 2009
%C A007047 Row sums of A038719. - _Peter Bala_, Jul 09 2014
%C A007047 Also the number of restricted barred preferential arrangements of an n-set having two bars, where one fixed section is a free section and the other two sections are restricted sections. - _Sithembele Nkonkobe_, Jun 16 2015
%C A007047 The number of all predictable outcomes of a race between a given number registered competitors, where clean finishes, dead heats (ties), disqualifications, cancellations and their combinations are all counted. (11 outcomes for two competitors, 51 for three, 299 for four, etc.. Example for two competitors shown below.) - _Gergely Földvári_, Jul 28 2024
%D A007047 V. Murali, Counting fuzzy subsets of a finite set, preprint, Rhodes University, Grahamstown 6140, South Africa, 2003.
%D A007047 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007047 Alois P. Heinz, <a href="/A007047/b007047.txt">Table of n, a(n) for n = 0..424</a> (first 101 terms from T. D. Noe)
%H A007047 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Barry4/barry271.html">General Eulerian Polynomials as Moments Using Exponential Riordan Arrays</a>, Journal of Integer Sequences, 16 (2013), #13.9.6.
%H A007047 Lisa Berry, Stefan Forcey, Maria Ronco, and Patrick Showers, <a href="http://www.math.uakron.edu/~sf34/stello_both.pdf">Species substitution, graph suspension, and graded hopf algebras of painted tree polytopes</a>.
%H A007047 Gergely Földvári, the total number of all predictable outcomes of a race between "k" number of registered competitors: <a href="/A007047/a007047.jpg">Race Outcome Formula</a>
%H A007047 Jun Ma, S.-M. Ma, and Y.-N. Yeh, <a href="https://arxiv.org/abs/1711.09016">Recurrence relations for binomial-Eulerian polynomials</a>, arXiv preprint arXiv:1711.09016 [math.CO], 2017.
%H A007047 T. Manneville and V. Pilaud, <a href="https://arxiv.org/abs/1501.07152">Compatibility fans for graphical nested complexes</a>, arXiv preprint arXiv:1501.07152 [math.CO], 2015-2016.
%H A007047 Rajesh Kumar Mohapatra and Tzung-Pei Hong, <a href="https://doi.org/10.3390/math10071161">On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences</a>, Mathematics (2022) Vol. 10, No. 7, 1161.
%H A007047 V. Murali, <a href="http://www.ru.ac.za/affiliates/fuzzysystems">Number of fuzzy subsets of a finite set</a>, fuzzy systems research group, Universities of Rhodes and Fort Hare. [broken link]
%H A007047 V. Murali and B. B. Makamba, <a href="https://doi.org/10.1080/03081070512331318356">Finite Fuzzy Sets</a>, International Journal of General Systems, Vol. 34, No. 1 (2005), pp. 61-75.
%H A007047 R. B. Nelsen, <a href="/A007047/a007047.pdf">Letter to N. J. A. Sloane, Aug. 1991</a>
%H A007047 R. B. Nelsen and H. Schmidt, Jr., <a href="http://www.jstor.org/stable/2690450">Chains in power sets</a>, Math. Mag., 64 (1991), 23-31.
%H A007047 S. Nkonkobe and V. Murali, <a href="https://arxiv.org/abs/1503.06172">A study of a family of generating functions of Nelsen-Schmidt type and some identities on restricted barred preferential arrangements</a>, arXiv:1503.06172 [math.CO], Apr 2015.
%H A007047 <a href="/index/Pos#posets">Index entries for sequences related to posets</a>
%F A007047 E.g.f.: exp(2*x)/(2-exp(x)).
%F A007047 a(n) = Sum_{k>=1} (k+1)^n/2^k = 2*A000629(n)-1. - _Benoit Cloitre_, Sep 08 2002
%F A007047 a(n) = one less than sum of quotients with numerator 4 times (n!)((k_1 + k_2 + ... + k_n)!) and with denominator (k_1!k_2!...k_n!)(1!^k_1 2!^k_2...n!^k_n) where the sum is taken over all partitions 1*k_1 + 2*k_2 + ... + n*k_n = n. E.g. a(1) = 3 because the membership value of x to {x} is either 1, alpha with 0 < alpha < 1 or 0. a(2) = 11 since the membership values x and y to {x, y} are 1 >= alpha >= beta >= 0 for {empty set, x, y} in that order or {empty set, y, x} exercising all possible > or = . - Venkat Murali (v.murali(AT)ru.ac.za), May 18 2005
%F A007047 G.f.: 1/Q(0), where Q(k) = 1 - 3*x*(k+1) - 2*x^2*(k+1)*(k+1)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Oct 02 2013
%F A007047 a(n) ~ 2*n! / (log(2))^(n+1). - _Vaclav Kotesovec_, Sep 27 2017
%F A007047 a(n) = 4 * A000670(n) - 1 for n > 0. - _Alois P. Heinz_, Feb 07 2020
%F A007047 a(n) = -(-1)^n Phi(2,-n,-1), where Phi(z,s,a) is the Lerch zeta function. - _Federico Provvedi_, Sep 05 2020
%F A007047 a(n) = 1 + 2 * Sum_{k=0..n-1} (-1)^(n-k-1) * binomial(n,k) * a(k). - _Ilya Gutkovskiy_, Apr 28 2021
%F A007047 From _Rajesh Kumar Mohapatra_, Jul 02 2025: (Start)
%F A007047 a(n) = 2*A000629(n) - 1.
%F A007047 a(n) = 2^n + Sum_{k=0..n-1} binomial(n,k) * a(k).
%F A007047 a(n) = 2^n + Sum_{k=1..n} binomial(n,k) * a(n-k), a(0) = 1. (End)
%e A007047 If there are two registered competitors, A and B, in a race, the total number of predictable outcomes counting all possibilities of clean finishes (f), dead heats (t), disqualifications (d), cancellations (c) and their combinations is 11 (eleven). Here is the breakdown: AfBf, BfAf, AtBt, AfBd, AfBc, BfAd, BfAc, AdBd, AcBc, AdBc, AcBd. - _Gergely Földvári_, Jul 28 2024
%p A007047 P := proc(n,x) option remember; if n = 0 then 1 else
%p A007047    (n*x+2*(1-x))*P(n-1,x)+x*(1-x)*diff(P(n-1,x),x);
%p A007047    expand(%) fi end:
%p A007047 A007047 := n -> 2^n*subs(x=1/2, P(n,x)):
%p A007047 seq(A007047(n), n=0..19);  # _Peter Luschny_, Mar 07 2014
%p A007047 # second Maple program:
%p A007047 b:= proc(n) option remember; `if`(n=0, 4,
%p A007047       add(b(n-j)*binomial(n, j), j=1..n))
%p A007047     end:
%p A007047 a:= n-> `if`(n=0, 1, b(n)-1):
%p A007047 seq(a(n), n=0..21);  # _Alois P. Heinz_, Feb 07 2020
%t A007047 Table[LerchPhi[1/2, -n, 2]/2, {n, 0, 10}] (* _Vladimir Reshetnikov_, Feb 16 2011 *)
%t A007047 Table[2*PolyLog[-n, 1/2] - 1 , {n, 0, 19}] (* _Jean-François Alcover_, Aug 14 2013 *)
%t A007047 With[{nn=20},CoefficientList[Series[Exp[2x]/(2-Exp[x]),{x,0,nn}],x] Range[ 0,nn]!] (* _Harvey P. Dale_, Dec 08 2015 *)
%t A007047 Table[-(-1)^k HurwitzLerchPhi[2, -k, -1], {k, 0, 30}] (* _Federico Provvedi_,Sep 05 2020 *)
%t A007047 a[n_]:= a[n] = 2^n + Sum[Binomial[n, k]*a[k], {k, 0, n-1}]; Table[a[n], {n, 0, 20}] (* _Rajesh Kumar Mohapatra_, Jul 02 2025 *)
%t A007047 a[0] = 1; a[n_]:= a[n] = 2^n + Sum[Binomial[n, k]*a[n-k], {k, 1, n}]; Table[a[n], {n, 0, 20}] (* _Rajesh Kumar Mohapatra_, Jul 02 2025 *)
%o A007047 (PARI) a(n)=if(n<0,0,n!*polcoeff(subst((y+1)^2/(1-y),y,exp(x+x*O(x^n))-1),n));
%o A007047 (PARI) my(x='x+O('x^66)); Vec(serlaplace(exp(2*x)/(2-exp(x)))) \\ _Joerg Arndt_, Aug 14 2013
%o A007047 (Haskell)
%o A007047 a007047 = sum . a038719_row  -- _Reinhard Zumkeller_, Feb 05 2014
%Y A007047 Cf. A000629, A000629, A000670. Row sums of A038719.
%K A007047 nonn,nice,easy
%O A007047 0,2
%A A007047 _N. J. A. Sloane_, Roger B. Nelsen