A005321 -id:A005321 - cp's OEIS frontend

A005327 Certain subgraphs of a directed graph (inverse binomial transform of A005321).

1, 0, 1, 6, 91, 2820, 177661, 22562946, 5753551231, 2940064679040, 3007686166657921, 6156733583148764286, 25211824022994189751171, 206510050572345408251841660, 3383254158526734823389921915781

Offset: 1

N. J. A. Sloane

nonn

q-Subfactorial for q=2. - Vladimir Reshetnikov, Sep 12 2016

T. L. Greenough, Enumeration of interval orders without duplicated holdings, Preprint, circa 1976.
T. L. Greenough and T. Lockman, Representation and enumeration of interval orders and semiorders, Ph.D. Thesis, Dartmouth, 1976.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 1..50
E. Andresen and K. Kjeldsen, On certain subgraphs of a complete transitively directed graph, Discrete Math. 14 (1976), no. 2, 103-119.
T. L. Greenough, Enumeration of interval orders without duplicated holdings, Preprint, circa 1976. [Annotated scanned copy]
Hsien-Kuei Hwang, Emma Yu Jin, and Michael J. Schlosser, Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow, arXiv:2012.13570 [math.CO], 2020.
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Subfactorial, q-Factorial, q-Analog.

Cf. A002820, A005329, A005321.

p:=proc(n) if n=0 then 1 else product(2^i-1,i=1..n) fi end: a:=n->p(n-1)*sum((-1)^j/p(j),j=0..n-1): seq(a(n),n=1..17); # Emeric Deutsch, Jan 23 2005

a[1] = 1; a[n_] := a[n] = (2^(n-1)-1)*a[n-1] + (-1)^(n-1); Array[a, 15] (* Jean-François Alcover, Apr 05 2016, after Max Alekseyev *)
With[{q = 2}, Table[QFactorial[n, q] Sum[(-1)^k/QFactorial[k, q], {k, 0, n}], {n, 0, 20}]] (* Vladimir Reshetnikov, Sep 12 2016 *)

For n>1, a(n) = (2^(n-1)-1)*a(n-1) + (-1)^(n-1). - Max Alekseyev, Feb 23 2012
a(n) = p(n-1)*sum((-1)^j/p(j), j=0..n-1), where p(0) = 1, p(k) = product(2^i-1, i=1..k) for k>0. - Emeric Deutsch, Jan 23 2005
a(n) ~ A048651^2 * 2^(n*(n-1)/2). - Vaclav Kotesovec, Oct 09 2019

More terms from Max Alekseyev, Apr 27 2010

A005014 Certain subgraphs of a directed graph (inverse binomial transform of A005321).

Original entry on oeis.org

1, 1, 7, 97, 2911, 180481, 22740607, 5776114177, 2945818230271, 3010626231336961, 6159741269315422207, 25217980756577338515457, 206535262396368402441592831, 3383460668577307168798173757441

Offset: 1

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Michael De Vlieger, Table of n, a(n) for n = 1..81
E. Andresen and K. Kjeldsen, On certain subgraphs of a complete transitively directed graph, Discrete Math. 14 (1976), no. 2, 103-119.
Hsien-Kuei Hwang, Emma Yu Jin, and Michael J. Schlosser, Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow, arXiv:2012.13570 [math.CO], 2020.
N. J. A. Sloane, Transforms

Pairwise sums of A005327.

p:=proc(n) if n=0 then 1 else product(2^i-1,i=1..n) fi end: a:=n->(-1)^n+(p(n)+p(n-1))*sum((-1)^j/p(j),j=0..n-1): seq(a(n),n=1..14); # Emeric Deutsch, Jan 23 2005

a[1] = 1; a[n_] := a[n] = (2^n-2)*a[n-1]-(-1)^n; Table[a[n], {n, 1, 14}] (* Jean-François Alcover, Jan 17 2014, after Vladeta Jovovic *)

a(n) = (-1)^n + (p(n) + p(n-1))Sum_{j=0..n-1} (-1)^j/p(j), where p(0)=1, p(k) = Product_{i=1..k} (2^i - 1) for k > 0. - Emeric Deutsch, Jan 23 2005
a(n) = (2^n-2)*a(n-1) - (-1)^n. - Vladeta Jovovic, Aug 20 2006
G.f.: Sum_{n>=0} (x^n*Product_{i=1..n} (2^i - 1)/(1 + 2^i*x)). - Vladeta Jovovic, Mar 10 2008

More terms from Vladeta Jovovic, Aug 20 2006

A005331 Certain subgraphs of a directed graph (binomial transform of A005321).

Original entry on oeis.org

1, 2, 5, 20, 179, 4082, 218225, 25316720, 6135834479, 3047003143022, 3067545380897645, 6223557209578656620, 25360384878802358268779, 207167485813280961035481962, 3389045635023473628621934703465, 110960673463328354866093662279119720, 7268699514166911556909103208203294815079

Offset: 0

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Michael De Vlieger, Table of n, a(n) for n = 0..81
E. Andresen and K. Kjeldsen, On certain subgraphs of a complete transitively directed graph, Discrete Math. 14 (1976), no. 2, 103-119.
Hsien-Kuei Hwang, Emma Yu Jin, and Michael J. Schlosser, Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow, arXiv:2012.13570 [math.CO], 2020.
N. J. A. Sloane, Transforms

Block[{nn = 16, s}, s = CoefficientList[Series[Sum[x^n*Product[(2^i - 1)/(1 + (2^i - 1)*x), {i, n}], {n, 0, #}], {x, 0, #}], x] &@ nn; Array[Sum[Binomial[#, j] s[[j + 1]], {j, 0, #}] &, nn + 1, 0]] (* Michael De Vlieger, Apr 13 2021, after Jean-François Alcover at A005321 *)

a(n) = Sum_{j=0..n} binomial(n,j)*A005321(j).

More terms from Max Alekseyev, May 04 2010

a(15)-a(16) from Michael De Vlieger, Apr 13 2021

A005329 a(n) = Product_{i=1..n} (2^i - 1). Also called 2-factorial numbers.

Original entry on oeis.org

1, 1, 3, 21, 315, 9765, 615195, 78129765, 19923090075, 10180699028325, 10414855105976475, 21319208401933844325, 87302158405919092510875, 715091979502883286756577125, 11715351900195736886933003038875, 383876935713713710574133710574817125

Offset: 0

N. J. A. Sloane

Conjecture: this sequence is the inverse binomial transform of A075272 or, equivalently, the inverse binomial transform of the BinomialMean transform of A075271. - John W. Layman, Sep 12 2002
To win a game, you must flip n+1 heads in a row, where n is the total number of tails flipped so far. Then the probability of winning for the first time after n tails is A005329 / A006125. The probability of having won before n+1 tails is A114604 / A006125. - Joshua Zucker, Dec 14 2005
Number of upper triangular n X n (0,1)-matrices with no zero rows. - Vladeta Jovovic, Mar 10 2008
Equals the q-Fibonacci series for q = (-2), and the series prefaced with a 1: (1, 1, 1, 3, 21, ...) dot (1, -2, 4, -8, ...) if n is even, and (-1, 2, -4, 8, ...) if n is odd. For example, a(3) = 21 = (1, 1, 1, 3) dot (-1, 2, -4, 8) = (-1, 2, -4, 24) and a(4) = 315 = (1, 1, 1, 3, 21) dot (1, -2, 4, -8 16) = (1, -2, 4, -24, 336). - Gary W. Adamson, Apr 17 2009
Number of chambers in an A_n(K) building where K=GF(2) is the field of two elements. This is also the number of maximal flags in an n-dimensional vector space over a field of two elements. - Marcos Spreafico, Mar 22 2012
Given probability p = 1/2^n that an outcome will occur at the n-th stage of an infinite process, then starting at n=1, A114604(n)/A006125(n+2) = 1-a(n)/A006125(n+1) is the probability that the outcome has occurred up to and including the n-th iteration. The limiting ratio is 1-A048651 ~ 0.7112119. These observations are a more formal and generalized statement of Joshua Zucker's Dec 14, 2005 comment. - Bob Selcoe, Mar 02 2016
Also the number of dominating sets in the n-triangular honeycomb rook graph. - Eric W. Weisstein, Jul 14 2017
Empirical: Letting Q denote the Hall-Littlewood Q basis of the symmetric functions over the field of fractions of the univariate polynomial ring in t over the field of rational numbers, and letting h denote the complete homogeneous basis, a(n) is equal to the absolute value of 2^A000292(n) times the coefficient of h_{1^(n*(n+1)/2)} in Q_{(n, n-1, ..., 1)} with t evaluated at 1/2. - John M. Campbell, Apr 30 2018
The series f(x) = Sum_{n>=0} x^(2^n-1)/a(n) satisfies f'(x) = f(x^2), f(0) = 1. - Lucas Larsen, Jan 05 2022

			G.f. = 1 + x + 3*x^2 + 21*x^3 + 315*x^4 + 9765*x^5 + 615195*x^6 + 78129765*x^7 + ...

Annie Cuyt, Vigdis Brevik Petersen, Brigitte Verdonk, Haakon Waadeland, and William B. Jones, Handbook of continued fractions for special functions, Springer, New York, 2008. (see 19.2.1)
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 358.
Mark Ronan, Lectures on Buildings (Perspectives in Mathematics; Vol. 7), Academic Press Inc., 1989.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..50
E. Andresen and K. Kjeldsen, On certain subgraphs of a complete transitively directed graph, Discrete Math., Vol. 14, No. 2 (1976), pp. 103-119.
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Hsien-Kuei Hwang, Emma Yu Jin, and Michael J. Schlosser, Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow, arXiv:2012.13570 [math.CO], 2020.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
Eric Weisstein's World of Mathematics, Dominating Set.
Eric Weisstein's World of Mathematics, q-Factorial.
Index entries for sequences related to factorial numbers.

Cf. A000225, A005321, A006125, A114604, A006088, A028362, A027871 (3-fac), A027872 (5-fac), A027873 (6-fac), A048651, A048652, A075271, A075272, A032085, A122746.
Cf. A048651, A079555, A152476 (inverse binomial transform).
Column q=2 of A069777.

List([0..15],n->Product([1..n],i->2^i-1)); # Muniru A Asiru, May 18 2018

[1] cat [&*[ 2^k-1: k in [1..n] ]: n in [1..16]]; // Vincenzo Librandi, Dec 24 2015

A005329 := proc(n) option remember; if n<=1 then 1 else (2^n-1)*procname(n-1); end if; end proc: seq(A005329(n), n=0..15);

a[0] = 1; a[n_] := a[n] = (2^n-1)*a[n-1]; a /@ Range[0,14] (* Jean-François Alcover, Apr 22 2011 *)
FoldList[Times, 1, 2^Range[15] - 1] (* Harvey P. Dale, Dec 21 2011 *)
Table[QFactorial[n, 2], {n, 0, 14}] (* Arkadiusz Wesolowski, Oct 30 2012 *)
QFactorial[Range[0, 10], 2] (* Eric W. Weisstein, Jul 14 2017 *)
a[ n_] := If[ n < 0, 0, (-1)^n QPochhammer[ 2, 2, n]]; (* Michael Somos, Jan 28 2018 *)

a(n)=polcoeff(sum(m=0,n,2^(m*(m+1)/2)*x^m/prod(k=0,m,1+2^k*x+x*O(x^n))),n) \\ Paul D. Hanna, Sep 17 2009

Dx(n,F)=local(D=F);for(i=1,n,D=deriv(D));D
a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+sum(k=1,n,x^k/k!*Dx(k,x*A+x*O(x^n) ))); polcoeff(A,n) \\ Paul D. Hanna, Apr 21 2012

{a(n) = if( n<0, 0, prod(k=1, n, 2^k - 1))}; /* Michael Somos, Jan 28 2018 */

{a(n) = if( n<0, 0, (-1)^n * sum(k=0, n+1, (-1)^k * 2^(k*(k+1)/2) * prod(j=1, k, (2^(n+1-j) - 1) / (2^j - 1))))}; /* Michael Somos, Jan 28 2018 */

a(n)/2^(n*(n+1)/2) -> c = 0.2887880950866024212788997219294585937270... (see A048651, A048652).
From Paul D. Hanna, Sep 17 2009: (Start)
G.f.: Sum_{n>=0} 2^(n*(n+1)/2) * x^n / (Product_{k=0..n} (1+2^k*x)).
Compare to: 1 = Sum_{n>=0} 2^(n*(n+1)/2) * x^n/(Product_{k=1..n+1} (1+2^k*x)). (End)
G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^n/n! * d^n/dx^n x*A(x). - Paul D. Hanna, Apr 21 2012
a(n) = 2^(binomial(n+1,2))*(1/2; 1/2){n}, where (a;q){n} is the q-Pochhammer symbol. - G. C. Greubel, Dec 23 2015
a(n) = Product_{i=1..n} A000225(i). - Michel Marcus, Dec 27 2015
From Peter Bala, Nov 10 2017: (Start)
O.g.f. as a continued fraction of Stieltjes' type: A(x) = 1/(1 - x/(1 - 2*x/(1 - 6*x/(1 - 12*x/(1 - 28*x/(1 - 56*x/(1 - ... -(2^n - 2^floor(n/2))*x/(1 - ... )))))))) (follows from Heine's continued fraction for the ratio of two q-hypergeometric series at q = 2. See Cuyt et al. 19.2.1).
A(x) = 1/(1 + x - 2*x/(1 - (2 - 1)^2*x/(1 + x - 2^3*x/(1 - (2^2 - 1)^2*x/(1 + x - 2^5*x/(1 - (2^3 - 1)^2*x/(1 + x - 2^7*x/(1 - (2^4 - 1)^2*x/(1 + x - ... ))))))))). (End)
0 = a(n)*(a(n+1) - a(n+2)) + 2*a(n+1)^2 for all n>=0. - Michael Somos, Feb 23 2019
From Amiram Eldar, Feb 19 2022: (Start)
Sum_{n>=0} 1/a(n) = A079555.
Sum_{n>=0} (-1)^n/a(n) = A048651. (End)

Better definition from Leslie Ann Goldberg (leslie(AT)dcs.warwick.ac.uk), Dec 11 1999

A002845 Number of distinct values taken by 2^2^...^2 (with n 2's and parentheses inserted in all possible ways).

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 17, 36, 78, 171, 379, 851, 1928, 4396, 10087, 23273, 53948, 125608, 293543, 688366, 1619087, 3818818, 9029719, 21400706, 50828664, 120963298, 288405081, 688821573, 1647853491, 3948189131, 9473431479

Offset: 1

N. J. A. Sloane

a(n) <= A002955(n). - Max Alekseyev, Sep 23 2009

			From _M. F. Hasler_, Apr 17 2024: (Start)
The table with explicit lists of values starts as follows:
   n | distinct values of 2^...^2 with all possible parenthesizations
-----+---------------------------------------------------------------
   1 | 2
   2 | 2^2 = 4
   3 | (2^2)^2 = 2^(2^2) = 16
   4 | (2^2^2)^2 = 2^8 = 256, (2^2)^(2^2) = 2^(2^2^2) = 2^16 (= 65536)
   5 | 256^2 = 2^16, (2^16)^2 = 2^32, 2^256, 2^2^16 (~ 2*10^19728)
   6 | (2^16)^2 = 2^32, 2^64, 2^512, 2^2^16, 2^2^17, 2^2^32, 2^2^256, 2^2^2^16
   7 | 2^64, 2^128, 2^256, 2^1024, 2^2^17, 2^2^18, 2^2^32, 2^2^33, 2^2^64, 2^2^257,
     | 2^2^512, 2^2^2^16, 2^2^65537, 2^2^2^17, 2^2^2^32, 2^2^2^256, 2^2^2^2^16
  ...| ...
(When parentheses are omitted above, we use that ^ is right associative.) (End)

J. Q. Longyear, personal communication.
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).

F. Goebel and R. P. Nederpelt, The number of numerical outcomes of iterated powers, Amer. Math. Monthly, 80 (1971), 1097-1103.
R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: front, back [Annotated scanned copy, with permission]
R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis, Amer. Math. Monthly 80 (8) (1973), 868-876.
R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis (annotated cached copy, with permission)
Sean A. Irvine, Java program (github).
J. Longyear, T. Trotter, and N. J. A. Sloane, Correspondence.
MathOverflow, MathOverflow discussion of related questions.
Vladimir Reshetnikov, Computes terms of OEIS sequence A002845 (C# library).
Jon E. Schoenfield, The 851 values for n=12.
Index entries for sequences related to parenthesizing

Cf. A003018, A003019, A145545, A145546, A145547, A145548, A145549, A145550, A000081.

/* Define operators for numbers represented (recursively) as list of positions of bits 1. Illustration using the commands below: T = 3.bits; T.int */
n.bits = vector(hammingweight(n), v,  n -= 1 << v= valuation(n, 2); v.bits)
ONE = 1.bits; m.int = sum(i=1, #m, 1<=0])}
{ADD(m, n, a=#m, b=#n)= if(!a, n, !b, m, a=b=1; until(a>#m|| b>#n, if(m[a]==n[b], until(a>=#m|| m[a]!=m[a+1]|| !#m=m[^a], m[a]=ADD(m[a],ONE)); b++, CMP(m[a], n[b])<0, a++, m=concat([m[1..a-1], [n[b]], m[a..#m]]); b++)); b>#n|| m=concat(m,n[b..#n]); m)}
{CMP(m, n, a=#m, b=#n, c=0)= if(!b, a, !a, -1, while(!(c=CMP(m[a], n[b]))&& a--&& b--, ); if(c, c, 1-b))}
{SUB(m, n, a=#n)= if(!a, m, my(b=a=1, c, i); while(a<=#m && b<=#n, if(0>c=CMP(m[a], n[b]), a++, c, i=[c=n[b]]; b++; while(m[a]!=c=ADD(c, ONE), if(b<=#n && c==n[b], b++, i=concat(i, [c]))); m=concat([m[1..a-1], i, m[a+1..#m]]); a += #i, m=m[^a]; b++)); m)}
A2845 = List([[2.bits]]) /* List of values for each n */
{A002845(n)= while(#A2845= 15. - M. F. Hasler, Apr 28 2024

a(12)-a(13) corrected and a(14)-a(27) added by Jon E. Schoenfield, Oct 11 2008
a(28)-a(29) computed by Kirill Osenkov, added by Vladimir Reshetnikov, Feb 07 2019
a(30)-a(31) added by Sean A. Irvine, Feb 18 2019

A138265 Number of upper triangular zero-one matrices with n ones and no zero rows or columns.

Original entry on oeis.org

1, 1, 1, 2, 5, 16, 61, 271, 1372, 7795, 49093, 339386, 2554596, 20794982, 182010945, 1704439030, 17003262470, 180011279335, 2015683264820, 23801055350435, 295563725628564, 3850618520827590, 52514066450469255, 748191494586458700, 11115833059268126770

Offset: 0

Vladeta Jovovic, Mar 10 2008, Mar 11 2008

Row sums of A193357.
This is also the number of rigid unlabeled interval orders with n points (see Brightwell-Keller, Theorem 2; or Dukes-Kitaev-Remmel-Steingrímsson, Theorem 8). - N. J. A. Sloane, Dec 04 2011  [Corrected by Vít Jelínek, Sep 04 2014.]
Number of length-n ascent sequences without flat steps (i.e., no two adjacent digits are equal).  An ascent sequence is a sequence [d(1), d(2), ..., d(n)] where d(k)>=0 and d(k) <= 1 + asc([d(1), d(2), ..., d(k-1)]) and asc(.) gives the number of ascents of its argument. [Joerg Arndt, Nov 05 2012]

			From _Joerg Arndt_, Nov 05 2012: (Start)
The a(4) = 5 such matrices with 4 ones are (dots for zeros):
  1 . . .      1 1 .      1 . 1      1 1 .      1 . .
  . 1 . .      . . 1      . 1 .      . 1 .      . 1 1
  . . 1 .      . . 1      . . 1      . . 1      . . 1
  . . . 1
The a(5)=16 ascent sequences without flat steps are (dots for zeros):
  [ 1]   [ . 1 . 1 . ]
  [ 2]   [ . 1 . 1 2 ]
  [ 3]   [ . 1 . 1 3 ]
  [ 4]   [ . 1 . 2 . ]
  [ 5]   [ . 1 . 2 1 ]
  [ 6]   [ . 1 . 2 3 ]
  [ 7]   [ . 1 2 . 1 ]
  [ 8]   [ . 1 2 . 2 ]
  [ 9]   [ . 1 2 . 3 ]
  [10]   [ . 1 2 1 . ]
  [11]   [ . 1 2 1 2 ]
  [12]   [ . 1 2 1 3 ]
  [13]   [ . 1 2 3 . ]
  [14]   [ . 1 2 3 1 ]
  [15]   [ . 1 2 3 2 ]
  [16]   [ . 1 2 3 4 ]
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..300
G. E. Andrews and J. A. Sellers, Congruences for the Fishburn Numbers, arXiv preprint arXiv:1401.5345 [math.NT], 2014 (see final paragraph of text).
George E. Andrews and Vít Jelínek, On q-Series Identities Related to Interval Orders, arXiv:1309.6669 [math.CO], 2013.
George E. Andrews and Vít Jelínek, On q-Series Identities Related to Interval Orders, European Journal of Combinatorics, Volume 39, July 2014, 178-187.
Graham Brightwell and Mitchel T. Keller, Asymptotic Enumeration of Labelled Interval Orders, arXiv 1111.6766 [math.CO], 2011.
Kathrin Bringmann, Yingkun Li, and Robert C. Rhoades, Asymptotics for the number of row-Fishburn matrices, European Journal of Combinatorics, vol.41, pp.183-196, (October-2014); preprint.
Matthieu Dien, Antoine Genitrini, and Frederic Peschanski, A Combinatorial Study of Async/Await Processes, Conf.: 19th Int'l Colloq. Theor. Aspects of Comp. (2022), (Analytic) Combinatorics of concurrent systems.
M. Dukes, S. Kitaev, J. Remmel, and E. Steingrímsson, Enumerating (2+2)-free posets by indistinguishable elements, arXiv:1006.2696 [math.CO], 2010.
M. Dukes, S. Kitaev, J. Remmel, and E. Steingrímsson, Enumerating (2+2)-free posets by indistinguishable elements, Journal of Combinatorics, Volume 2, Issue 1, 2011, pp. 139-163.
F. Garvan, Congruences and relations for the r-Fishburn numbers, arXiv:1406.5611 [math.NT], 2014.
Ankush Goswami, Abhash Kumar Jha, Byungchan Kim, and Robert Osburn, Asymptotics and sign patterns for coefficients in expansions of Habiro elements, arXiv:2204.02628 [math.NT], 2022.
Hsien-Kuei Hwang and Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.
V. Jelínek, Counting self-dual interval orders, arXiv:1106.2261 [math.CO], 2011.
V. Jelínek, Counting general and self-dual interval orders, Journal of Combinatorial Theory, Series A, Volume 119, Issue 3, April 2012, pp. 599-614.
V. Jelinek, Catalan pairs and Fishburn triples, Adv. Appl. Math. 70 (2015) 1-31
Soheir Mohamed Khamis, Exact Counting of Unlabeled Rigid Interval Posets Regarding or Disregarding Height, Order (journal), published on-line, 2011.
Yan X. Zhang, Four Variations on Graded Posets, arXiv preprint arXiv:1508.00318 [math.CO], 2015.

Cf. A005321, A104602, A135588, A193548, A194530.
Column k=0 of A242153.
Column k=1 of A264909.
Row sums of A137252.

g:=sum(product(1-1/(1+x)^i,i=1..n),n=0..35): gser:=series(g,x=0,30): seq(coeff(gser,x,n),n=0..22);  # Emeric Deutsch, Mar 23 2008
# second Maple program:
b:= proc(n, i, t) option remember; `if`(n<1, 1, add(
     `if`(i=j, 0, b(n-1, j, t+`if`(j>i, 1, 0))), j=0..t+1))
    end:
a:= n-> b(n-1, 0$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 09 2012, Jan 14 2015

max = 25; g = Sum[Product[1 - 1/(1 - x)^i, {i, 1, n}], {n, 0, max}]; gser = Series[g, {x, 0, max}]; a[n_] := SeriesCoefficient[gser, {x, 0, n}]; Table[a[n] // Abs, {n, 0, max-1}] (* Jean-François Alcover, Jan 24 2014, after Emeric Deutsch *)

# Adaptation of the Maple program by Alois P. Heinz:
@CachedFunction
def b(n, i, t):
    if n<1: return 1
    return sum(b(n-1, j, t+(j>i)) for j in range(t+2))
def a(n):
    if n<1: return 1
    return sum((-1)^(n-k)*binomial(n-1, k-1)*b(k-1, 0, 0) for k in range(n+1))
[a(n) for n in range(33)]
# Joerg Arndt, Feb 26 2014

G.f.: Sum_{n>=0} (Product_{i=1..n} 1-1/(1+x)^i).
G.f.: Sum_{n>=0} (1+x)^(n+1)*Product_{i=1..n} (1-(1+x)^i)^2. Proved by Bringmann-Li-Rhoades, and by Andrews-Jelínek. - Vít Jelínek, Sep 04 2014
a(n) = (1/n!)*Sum_{k=0..n} Stirling1(n,k)*A079144(k). a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n-1,k-1)*A022493(k).
G.f.: B(x/(1+x)) where B(x) is the g.f. of A022493; g.f.: Q(0,u) where u=x/(1+x), Q(k,u) = 1 + (1 - (1-x)^(2*k+1))/(1 - (1-(1-x)^(2*k+2))/(1 -(1-x)^(2*k+2) + 1/Q(k+1,u) )); (continued fraction). - Sergei N. Gladkovskii, Oct 03 2013
Asymptotics (Brightwell and Keller, 2011): a(n) ~ 12*sqrt(3)/(exp(Pi^2/12)*Pi^(5/2)) * n!*sqrt(n)*(6/Pi^2)^n. - Vaclav Kotesovec, May 03 2014
From Vít Jelínek, Sep 04 2014: (Start)
For each m, a(5m+4) mod 5 = 0. Conjectured by Andrews-Sellers, and proved by Garvan (see Remark 1.4(ii) in Garvan's paper).
For each m, a(5m+1) mod 5 = a(5m+2) mod 5 = 3*a(5m+3) mod 5. Proved by Garvan (see (1.17) in Garvan's paper).
The limit a(n)/A022493(n) is equal to exp(-Pi^2/6). This corresponds to the asymptotic probability that a random unlabeled interval order is rigid (See Brightwell-Keller; or Jelínek, Fact 5.2). (End)
Conjectural g.f.:  1 + Sum_{n >= 0} n/(1+x)^(n+1) * (Product_{i = 1..n} 1 - 1/(1+x)^i). Cf. A194530. - Peter Bala, Aug 21 2023

More terms from Emeric Deutsch, Mar 23 2008

A289314 Number of n X n Fishburn matrices with entries in the set {0,1,2}.

Original entry on oeis.org

1, 2, 12, 264, 19632, 4606752, 3311447232, 7202118117504, 47151987852663552, 927337336972381327872, 54741643544083873448266752, 9696222929066933463021344262144, 5152757080697434799933013959862300672, 8215035458438940398186389046297459974152192

Offset: 0

Peter Bala, Jul 03 2017

A Fishburn matrix is defined to be an upper-triangular matrix with nonnegative integer entries such that each row and column contains a nonzero entry. See A005321 for primitive Fishburn matrices of dimension n, that is, Fishburn matrices of dimension n with entries in the set {0,1}.

The present sequence has an alternative description as the number of primitive Fishburn matrices of dimension n where the 1's may be colored either black or white.

			a(2) = 12: The twelve 2 X 2 Fishburn matrices with entries 0, 1 or 2 are
/1 0\  /1 0\  /2 0\  /2 0\
\0 1/  \0 2/  \0 1/  \0 2/
/1 1\  /1 2\  /1 1\  /1 2\  /2 1\  /2 2\  /2 1\  /2 2\.
\0 1/  \0 1/  \0 2/  \0 2/  \0 1/  \0 1/  \0 2/  \0 2/
Alternatively, the twelve 2-colored primitive Fishburn matrices of dimension 2 (using +1 and -1 for the two different colored versions of 1) are
/+-1  0\ (4 possibilities)
\0  +-1/
   and
/+-1 +-1\ (8 possibilities).
\ 0  +-1/

Robert Israel, Table of n, a(n) for n = 0..64
Hsien-Kuei Hwang, Emma Yu Jin, and Michael J. Schlosser, Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow, arXiv:2012.13570 [math.CO], 2020.
Vít Jelínek, Counting general and self-dual interval orders, Journal of Combinatorial Theory, Series A, Volume 119, Issue 3, April 2012, pp. 599-614; arXiv preprint, arXiv:1106.2261 [math.CO], 2011.

Cf. A005321, A022493, A138265, A289315.

N:= 20: # to get a(0)..a(N)
g:= add(x^n*mul((3^i-1)/(1+x*(3^i-1)),i=1..n),n=0..N):
S:= series(g,x,N+1):
seq(coeff(S,x,j),j=0..N); # Robert Israel, Jul 11 2017

QP = QPochhammer; nmax = 14;
Sum[(-1)^n (1-x)^(-n-1) x^n QP[3, 3, n]/QP[x/(x-1), 3, n+1], {n, 0, nmax}] + O[x]^nmax // CoefficientList[#, x]& (* Jean-François Alcover, Sep 19 2018 *)

O.g.f.: A(x) = Sum_{n >=0} x^n Product_{i = 1..n} (3^i - 1)/(1 + x*(3^i - 1)) = 1 + 2*x + 12*x^2 + 264*x^3 + ... (use Jelínek, Theorem 2.1 with v = w = x = y = 2).
Two conjectural continued fractions for the o.g.f.:
A(x) = 1/(1 - 2*x/(1 - 4*x/(1 - 24*x/(1 - 64*x/(1 - 234*x/(1 - 676*x/(1 - ... - 3^(n-1)*(3^n - 1)*x/(1 - (3^n - 1)^2*x/(1 - ...))))))))) and
A(x) = 1 + 2*x/(1 - 6*x/(1 - 16*x/(1 - 72*x/(1 - 208*x/(1 - ... - 3^n*(3^n - 1)*x/(1 - (3^(n+1) - 1)*(3^n - 1)*x/(1 - ...))))))).
a(n) ~ c * 3^(n*(n+1)/2), where c = QPochhammer(1/3)^2 = 0.313741223174946734265526469975707962872482170305592991802056615373429729... - Vaclav Kotesovec, Aug 31 2023, updated Mar 17 2024

A289315 Number of n X n Fishburn matrices with entries in the set {0,1,2,3}.

Original entry on oeis.org

1, 3, 36, 2052, 505764, 511718148, 2088275673636, 34176650317115652, 2239082850356711468964, 586908388119824949146284548, 615402202729113953115253336166436, 2581165458211746544190089033131172341252, 43304685245392697816407075492352986194550240164

Offset: 0

Peter Bala, Jul 03 2017

The present sequence has an alternative description as the number of primitive Fishburn matrices of dimension n where each entry equal to 1 can have one of three different colors. Cf. A289314.

			a(2) = 36: The 36 2 X 2 Fishburn matrices with entries 0, 1, 2 or 3 are
/1 0\ /1 0\ /2 0\ /2 0\
\0 1/ \0 2/ \0 1/ \0 2/
/1 0\ /3 0\ /3 0\
\0 3/ \0 1/ \0 3/
/2 0\ /3 0\
\0 3/ \0 2/
/1 2\ /1 1\ /1 2\ /2 1\ /2 2\ /2 1\ /2 2\
\0 1/ \0 2/ \0 2/ \0 1/ \0 1/ \0 2/ \0 2/
/1 1\ /1 3\ /1 1\ /1 3\ /3 1\ /3 3\ /3 1\
\0 1/ \0 1/ \0 3/ \0 3/ \0 1/ \0 1/ \0 3/
/2 3\ /2 2\ /2 3\ /3 2\ /3 3\ /3 2\ /3 3\
\0 2/ \0 3/ \0 3/ \0 2/ \0 2/ \0 3/ \0 3/
/1 2\ /1 3\ /2 3\ /2 1\ /3 1\ /3 2\.
\0 3/ \0 2/ \0 1/ \0 3/ \0 2/ \0 1/
Alternatively, the 36 3-colored primitive Fishburn matrices of dimension 2 (using 1_j, j = 1,2,3 to denote the three different colored versions of 1) are
/1_j  0\ (9 possibilities)
\ 0 1_j/
and
/1_j 1_j\ (27 possibilities).
\ 0  1_j/

Robert Israel, Table of n, a(n) for n = 0..57
Hsien-Kuei Hwang, Emma Yu Jin, and Michael J. Schlosser, Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow, arXiv:2012.13570 [math.CO], 2020.
Vít Jelínek, Counting general and self-dual interval orders, Journal of Combinatorial Theory, Series A, Volume 119, Issue 3, April 2012, pp. 599-614; arXiv preprint, arXiv:1106.2261 [math.CO], 2011.

Cf. A005321, A022493, A138265, A289314.

G:= add(x^n*mul((4^i-1)/(1+x*(4^i-1)), i=1..n),n=0..40):
S:= series(G,x,41):
seq(coeff(S,x,j),j=0..40); # Robert Israel, Jul 10 2017

Table[SeriesCoefficient[Sum[x^j*Product[(4^i - 1)/(1 + x (4^i - 1)), {i, j}], {j, 0, n}], {x, 0, n}], {n, 0, 12}] (* Michael De Vlieger, Jul 10 2017, after Maple by Robert Israel *)

O.g.f.: A(x) = Sum_{n >= 0} x^n Product_{i = 1..n} (4^i - 1)/(1 + x*(4^i - 1)) = 1 + 3*x + 36*x^2 + 2052*x^3 + ... (use Jelínek, Theorem 2.1 with v = w = x = y = 3).
Two conjectural continued fractions for the o.g.f.:
A(x) = 1/(1 - 3*x/(1 - 9*x/(1 - 60*x/(1 - 225*x/(1 - 1008*x/(1 - 3969*x/(1 - ... - 4^(n-1)*(4^n - 1)*x/(1 - (4^n - 1)^2*x/(1 - ...))))))))) and
A(x) = 1 + 3*x/(1 - 12*x/(1 - 45*x/(1 - 240*x/(1 - 945*x/(1 - ... - 4^n*(4^n - 1)*x/(1 - (4^(n+1) - 1)*(4^n - 1)*x/(1 - ...))))))).
a(n) ~ c * 2^(n*(n+1)), where c = QPochhammer(1/4)^2 = 0.474083940023743191581900099468175063640311684514259231... - Vaclav Kotesovec, Aug 31 2023, updated Mar 17 2024

A137252 Triangle T(n,k) read by rows: number of k X k triangular (0,1)-matrices with exactly n entries equal to 1 and no zero rows or columns.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 4, 1, 0, 0, 0, 4, 11, 1, 0, 0, 0, 1, 33, 26, 1, 0, 0, 0, 0, 42, 171, 57, 1, 0, 0, 0, 0, 26, 507, 718, 120, 1, 0, 0, 0, 0, 8, 840, 4017, 2682, 247, 1, 0, 0, 0, 0, 1, 865, 12866, 25531, 9327, 502, 1, 0, 0, 0, 0, 0, 584, 26831, 138080, 141904, 30973, 1013, 1

Offset: 0

Vladeta Jovovic, Mar 11 2008

			Triangle T(n,k) begins:
  1;
  0, 1;
  0, 0, 1;
  0, 0, 1, 1;
  0, 0, 0, 4,  1;
  0, 0, 0, 4, 11,   1;
  0, 0, 0, 1, 33,  26,   1;
  0, 0, 0, 0, 42, 171,  57,   1;
  0, 0, 0, 0, 26, 507, 718, 120,  1;
  ...

Alois P. Heinz, Rows n = 0..100, flattened
Matthieu Dien, Antoine Genitrini, and Frederic Peschanski, A Combinatorial Study of Async/Await Processes, Conf.: 19th Int'l Colloq. Theor. Aspects of Comp. (2022), (Analytic) Combinatorics of concurrent systems.
M. Dukes, S. Kitaev, J. Remmel, E. Steingrimsson, Enumerating (2+2)-free posets by indistinguishable elements, J. Combin. 2 (1) (2011) 139-163 doi:10.4310/JOC.2011.v2.n1.a6, Figure 2; arXiv preprint arXiv:1006.2696 [math.CO], 2010-2011.

Cf. A138265 (row sums), A005321 (column sums), A135589.

T(2n,n) gives A357140.

G.f.: Sum(Product(1-1/(1+((1+x)^i-1)*y), i=1..n), n=0..infinity).

A339807 Irregular triangle read by rows: T(n,k) (n>=2, k>=1) is the number of strong digraphs on n nodes with k descents.

Original entry on oeis.org

1, 2, 11, 5, 10, 154, 540, 581, 272, 49, 122, 3418, 27304, 90277, 150948, 150519, 95088, 37797, 8714, 893, 3346, 142760, 1938178, 12186976, 42696630, 94605036, 145009210, 161845163, 134933733, 84656743, 39632149, 13481441, 3156845, 455917, 30649

Offset: 2

Hugo Pfoertner, Dec 28 2020

T(n,1) = A005321(n-1). Length of row n = binomial(n,2). It appears that T(n,binomial(n,2)) = A348901(n-1). - Geoffrey Critzer, Feb 12 2025

			Triangle begins:
 1;
 2, 11, 5;
 10, 154, 540, 581, 272, 49;
 122, 3418, 27304, 90277, 150948, 150519, 95088, 37797, 8714, 893;
 3346, 142760, 1938178, 12186976, 42696630, 94605036, 145009210, 161845163, 134933733, 84656743, 39632149, 13481441, 3156845, 455917, 30649;
 ...

Kassie Archer, Ira M. Gessel, Christina Graves, and Xuming Liang, Counting acyclic and strong digraphs by descents, arXiv:1909.01550 [math.CO], 20 Mar 2020.

Cf. A003030 (row sums), A057273 (another version of the same triangle), A307049, A339590, A005321, A000217.

nn = 8; B[n_] := FunctionExpand[QFactorial[n, (1 + u y)/(1 + y)]] (1 + y)^Binomial[n, 2]; g[z_] := Sum[(1 + u y)^Binomial[n, 2] z^n/FunctionExpand[QFactorial[n, (1 + u y)/(1 + y)]], {n, 0, nn}]; egf[eggf_] := Normal[Series[eggf, {z, 0, nn}]] /.Table[z^i -> z^i*B[i]/i!, {i, 1, nn + 1}]; Map[Drop[#, 1] &, Drop[Map[CoefficientList[#, u] &, Table[n!, {n, 0, nn}]CoefficientList[Series[-Log[egf[1/g[z]]], {z, 0, nn}], z] /. y -> 1], 2]] // Grid (* Geoffrey Critzer, Feb 12 2025 *)

Row 2 added by N. J. A. Sloane, Dec 29 2020

cp's OEIS Frontend

A005327 Certain subgraphs of a directed graph (inverse binomial transform of A005321).

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A005014 Certain subgraphs of a directed graph (inverse binomial transform of A005321).

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A005331 Certain subgraphs of a directed graph (binomial transform of A005321).

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A005329 a(n) = Product_{i=1..n} (2^i - 1). Also called 2-factorial numbers.

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A002845 Number of distinct values taken by 2^2^...^2 (with n 2's and parentheses inserted in all possible ways).

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A138265 Number of upper triangular zero-one matrices with n ones and no zero rows or columns.

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