A248925 -id:A248925 - cp's OEIS frontend

A011782 Coefficients of expansion of (1-x)/(1-2*x) in powers of x.

1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592

Offset: 0

Lee D. Killough (killough(AT)wagner.convex.com)

Apart from initial term, same as A000079 (powers of 2).
Number of compositions (ordered partitions) of n. - Toby Bartels, Aug 27 2003
Number of ways of putting n unlabeled items into (any number of) labeled boxes where every box contains at least one item. Also "unimodal permutations of n items", i.e., those which rise then fall. (E.g., for three items: ABC, ACB, BCA and CBA are unimodal.) - Henry Bottomley, Jan 17 2001
Number of permutations in S_n avoiding the patterns 213 and 312. - Tuwani Albert Tshifhumulo, Apr 20 2001. More generally (see Simion and Schmidt), the number of permutations in S_n avoiding (i) the 123 and 132 patterns; (ii) the 123 and 213 patterns; (iii) the 132 and 213 patterns; (iv) the 132 and 231 patterns; (v) the 132 and 312 patterns; (vi) the 213 and 231 patterns; (vii) the 213 and 312 patterns; (viii) the 231 and 312 patterns; (ix) the 231 and 321 patterns; (x) the 312 and 321 patterns.
a(n+2) is the number of distinct Boolean functions of n variables under action of symmetric group.
Number of unlabeled (1+2)-free posets. - Detlef Pauly, May 25 2003
Image of the central binomial coefficients A000984 under the Riordan array ((1-x), x*(1-x)). - Paul Barry, Mar 18 2005
Binomial transform of (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...); inverse binomial transform of A007051. - Philippe Deléham, Jul 04 2005
Also, number of rationals in [0, 1) whose binary expansions terminate after n bits. - Brad Chalfan, May 29 2006
Equals row sums of triangle A144157. - Gary W. Adamson, Sep 12 2008
Prepend A089067 with a 1, getting (1, 1, 3, 5, 13, 23, 51, ...) as polcoeff A(x); then (1, 1, 2, 4, 8, 16, ...) = A(x)/A(x^2). - Gary W. Adamson, Feb 18 2010
An elephant sequence, see A175655. For the central square four A[5] vectors, with decimal values 2, 8, 32 and 128, lead to this sequence. For the corner squares these vectors lead to the companion sequence A094373. - Johannes W. Meijer, Aug 15 2010
From Paul Curtz, Jul 20 2011: (Start)
Array T(m,n) = 2*T(m,n-1) + T(m-1,n):
  1, 1,  2,   4,   8,   16, ... = a(n)
  1, 3,  8,  20,  48,  112, ... = A001792,
  1, 5, 18,  56, 160,  432, ... = A001793,
  1, 7, 32, 120, 400, 1232, ... = A001794,
  1, 9, 50, 220, 840, 2912, ... = A006974, followed with A006975, A006976, gives nonzero coefficients of Chebyshev polynomials of first kind A039991 =
  1,
  1, 0,
  2, 0, -1,
  4, 0, -3, 0,
  8, 0, -8, 0, 1.
T(m,n) third vertical: 2*n^2, n positive (A001105).
Fourth vertical appears in Janet table even rows, last vertical (A168342 array, A138509, rank 3, 13, = A166911)). (End)
A131577(n) and differences are:
  0, 1, 2, 4, 8, 16,
  1, 1, 2, 4, 8, 16,  = a(n),
  0, 1, 2, 4, 8, 16,
  1, 1, 2, 4, 8, 16.
Number of 2-color necklaces of length 2n equal to their complemented reversal. For length 2n+1, the number is 0. - David W. Wilson, Jan 01 2012
Edges and also central terms of triangle A198069: a(0) = A198069(0,0) and for n > 0: a(n) = A198069(n,0) = A198069(n,2^n) = A198069(n,2^(n-1)). - Reinhard Zumkeller, May 26 2013
These could be called the composition numbers (see the second comment) since the equivalent sequence for partitions is A000041, the partition numbers. - Omar E. Pol, Aug 28 2013
Number of self conjugate integer partitions with exactly n parts for n>=1. - David Christopher, Aug 18 2014
The sequence is the INVERT transform of (1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...). - Gary W. Adamson, Jul 16 2015
Number of threshold graphs on n nodes [Hougardy]. - Falk Hüffner, Dec 03 2015
Number of ternary words of length n in which binary subwords appear in the form 10...0. - Milan Janjic, Jan 25 2017
a(n) is the number of words of length n over an alphabet of two letters, of which one letter appears an even number of times (the empty word of length 0 is included). See the analogous odd number case in A131577, and the Balakrishnan reference in A006516 (the 4-letter odd case), pp. 68-69, problems 2.66, 2.67 and 2.68. - Wolfdieter Lang, Jul 17 2017
Number of D-equivalence classes of Łukasiewicz paths. Łukasiewicz paths are D-equivalent iff the positions of pattern D are identical in these paths. - Sergey Kirgizov, Apr 08 2018
Number of color patterns (set partitions) for an oriented row of length n using two or fewer colors (subsets). Two color patterns are equivalent if we permute the colors. For a(4)=8, the 4 achiral patterns are AAAA, AABB, ABAB, and ABBA; the 4 chiral patterns are the 2 pairs AAAB-ABBB and AABA-ABAA. - Robert A. Russell, Oct 30 2018
The determinant of the symmetric n X n matrix M defined by M(i,j) = (-1)^max(i,j) for 1 <= i,j <= n is equal to a(n) * (-1)^(n*(n+1)/2). - Bernard Schott, Dec 29 2018
For n>=1, a(n) is the number of permutations of length n whose cyclic representations can be written in such a way that when the cycle parentheses are removed what remains is 1 through n in natural order. For example, a(4)=8 since there are exactly 8 permutations of this form, namely, (1 2 3 4), (1)(2 3 4), (1 2)(3 4), (1 2 3)(4), (1)(2)(3 4), (1)(2 3)(4), (1 2)(3)(4), and (1)(2)(3)(4). Our result follows readily by conditioning on k, the number of parentheses pairs of the form ")(" in the cyclic representation. Since there are C(n-1,k) ways to insert these in the cyclic representation and since k runs from 0 to n-1, we obtain a(n) = Sum_{k=0..n-1} C(n-1,k) = 2^(n-1). - Dennis P. Walsh, May 23 2020
Maximum number of preimages that a permutation of length n + 1 can have under the consecutive-231-avoiding stack-sorting map. - Colin Defant, Aug 28 2020
a(n) is the number of occurrences of the empty set {} in the von Neumann ordinals from 0 up to n. Each ordinal k is defined as the set of all smaller ordinals: 0 = {}, 1 = {0}, 2 = {0,1}, etc. Since {} is the foundational element of all ordinals, the total number of times it appears grows as powers of 2. - Kyle Wyonch, Mar 30 2025

			G.f. = 1 + x + 2*x^2 + 4*x^3 + 8*x^4 + 16*x^5 + 32*x^6 + 64*x^7 + 128*x^8 + ...
    ( -1   1  -1)
det (  1   1   1)  = 4
    ( -1  -1  -1)

Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education Journal, Vol. 31, No. 1, pp. 24-28, Winter 1997.
S. Kitaev, Patterns in Permutations and Words, Springer-Verlag, 2011. see p. 399 Table A.7
Xavier Merlin, Methodix Algèbre, Ellipses, 1995, p. 153.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Michael A. Allen, On a Two-Parameter Family of Generalizations of Pascal's Triangle, arXiv:2209.01377 [math.CO], 2022.
Christopher Bao, Yunseo Choi, Katelyn Gan, and Owen Zhang, On a Conjecture by Baril, Cerbai, Khalil, and Vajnovszki on Two Restricted Stacks, arXiv:2308.09344 [math.CO], 2023.
Jean-Luc Baril, Sergey Kirgizov and Armen Petrossian, Enumeration of Łukasiewicz paths modulo some patterns, arXiv:1804.01293 [math.CO], 2018.
Jean-Luc Baril, Sergey Kirgizov, and Vincent Vajnovszki, Descent distribution on Catalan words avoiding a pattern of length at most three, arXiv:1803.06706 [math.CO], 2018.
Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Christian Bean, Bjarki Gudmundsson, and Henning Ulfarsson, Automatic discovery of structural rules of permutation classes, arXiv:1705.04109 [math.CO], 2017.
Daniel Birmajer, Juan B. Gil, Jordan O. Tirrell, and Michael D. Weiner, Pattern-avoiding stabilized-interval-free permutations, arXiv:2306.03155 [math.CO], 2023.
Joshua P. Bowman, Compositions with an Odd Number of Parts, and Other Congruences, J. Int. Seq (2024) Vol. 27, Art. 24.3.6. See p. 14.
Giulio Cerbai, Anders Claesson, and Luca Ferrari, Stack sorting with restricted stacks, arXiv:1907.08142 [cs.DS], 2019.
Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015.
Johann Cigler, Recurrences for certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials, arXiv:2212.02118 [math.NT], 2022.
Bishal Deb, Cyclic sieving phenomena via combinatorics of continued fractions, arXiv:2508.13709 [math.CO], 2025. See p. 42.
Colin Defant and Kai Zheng, Stack-sorting with consecutive-pattern-avoiding stacks, arXiv:2008.12297 [math.CO], 2020.
John B. Dobson, A matrix variation on Ramus's identity for lacunary sums of binomial coefficients, arXiv preprint arXiv:1610.09361 [math.NT], 2016.
Mareike Fischer, Extremal Values of the Sackin Tree Balance Index, Ann. Comb. (2021) Vol. 25, 515-541, Remark 10.
Juan B. Gil and Jessica A. Tomasko, Fibonacci colored compositions and applications, arXiv:2108.06462 [math.CO], 2021.
Hannah Golab, Pattern avoidance in Cayley permutations, Master's Thesis, Northern Arizona Univ. (2024). See p. 25.
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 23.
Mats Granvik, Alternating powers of 2 as convoluted divisor recurrence
Nickolas Hein and Jia Huang, Variations of the Catalan numbers from some nonassociative binary operations, arXiv:1807.04623 [math.CO], 2018.
Nickolas Hein and Jia Huang, Modular Catalan Numbers, arXiv:1508.01688 [math.CO], 2015-2016. See Table 1.1 p. 2.
S. Heubach and T. Mansour, Counting rises, levels and drops in compositions, arXiv:math/0310197 [math.CO], 2003.
S. Hougardy, Classes of perfect graphs, Discr. Math. 306 (2006), 2529-2571.
Sergey Kitaev, Jeffrey Remmel and Mark Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv:1201.6243v1 [math.CO], 2012 (Corollary 3, case k=2, pages 10-11). - From _N. J. A. Sloane_, May 09 2012
Sergey Kitaev, Jeffrey Remmel, and Mark Tiefenbruck, Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II, Electronic Journal of Combinatorial Number Theory, Volume 15 #A16. Also on arXiv, arXiv:1302.2274 [math.CO], 2013.
Olivia Nabawanda and Fanja Rakotondrajao, The sets of flattened partitions with forbidden patterns, arXiv:2011.07304 [math.CO], 2020.
R. A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
L. Pudwell, Pattern avoidance in trees (slides from a talk, mentions many sequences), 2012. - From _N. J. A. Sloane_, Jan 03 2013
Santiago Rojas-Rojas, Camila Muñoz, Edgar Barriga, Pablo Solano, Aldo Delgado, and Carla Hermann-Avigliano, Analytic Evolution for Complex Coupled Tight-Binding Models: Applications to Quantum Light Manipulation, arXiv:2310.12366 [quant-ph], 2023. See p. 12.
R. Simion and F. W. Schmidt, Restricted permutations, European J. Combin., 6, 383-406, 1985, see pp. 392-393.
Christoph Wernhard and Wolfgang Bibel, Learning from Łukasiewicz and Meredith: Investigations into Proof Structures (Extended Version), arXiv:2104.13645 [cs.AI], 2021.
Yan X. Zhang, Four Variations on Graded Posets, arXiv preprint arXiv:1508.00318 [math.CO], 2015.
Index entries for sequences related to Boolean functions
Index to divisibility sequences
Index entries for related partition-counting sequences
Index entries for linear recurrences with constant coefficients, signature (2).
Index entries for sequences related to Chebyshev polynomials.

Cf. A000670, A051486, A053120, A057711, A080929, A080961, A082138.
Cf. A082139, A082140, A082141, A089067, A144157, A131577, A211216.
Sequences with g.f.'s of the form ((1-x)/(1-2*x))^k: this sequence (k=1), A045623 (k=2), A058396 (k=3), A062109 (k=4), A169792 (k=5), A169793 (k=6), A169794 (k=7), A169795 (k=8), A169796 (k=9), A169797 (k=10).
Cf. A005418 (unoriented), A122746(n-3) (chiral), A016116 (achiral).
Row sums of triangle A100257.
A row of A160232.
Row 2 of A278984.

a011782 n = a011782_list !! n
a011782_list = 1 : scanl1 (+) a011782_list
-- Reinhard Zumkeller, Jul 21 2013

[Floor((1+2^n)/2): n in [0..35]]; // Vincenzo Librandi, Aug 21 2011

A011782:= n-> ceil(2^(n-1)): seq(A011782(n), n=0..50); # Wesley Ivan Hurt, Feb 21 2015
with(PolynomialTools):  A011782:=seq(coeftayl((1-x)/(1-2*x), x = 0, k),k=0..10^2); # Muniru A Asiru, Sep 26 2017

f[s_] := Append[s, Ceiling[Plus @@ s]]; Nest[f, {1}, 32] (* Robert G. Wilson v, Jul 07 2006 *)
CoefficientList[ Series[(1-x)/(1-2x), {x, 0, 32}], x] (* Robert G. Wilson v, Jul 07 2006 *)
Table[Sum[StirlingS2[n, k], {k,0,2}], {n, 0, 30}] (* Robert A. Russell, Apr 25 2018 *)
Join[{1},NestList[2#&,1,40]] (* Harvey P. Dale, Dec 06 2018 *)

PARI
```
{a(n) = if( n<1, n==0, 2^(n-1))};
```

Vec((1-x)/(1-2*x) + O(x^30)) \\ Altug Alkan, Oct 31 2015

def A011782(n): return 1 if n == 0 else 2**(n-1) # Chai Wah Wu, May 11 2022

[sum(stirling_number2(n,j) for j in (0..2)) for n in (0..35)] # G. C. Greubel, Jun 02 2020

a(0) = 1, a(n) = 2^(n-1).
G.f.: (1 - x) / (1 - 2*x) = 1 / (1 - x / (1 - x)). - Michael Somos, Apr 18 2012
E.g.f.: cosh(z)*exp(z) = (exp(2*z) + 1)/2.
a(0) = 1 and for n>0, a(n) = sum of all previous terms.
a(n) = Sum_{k=0..n} binomial(n, 2*k). - Paul Barry, Feb 25 2003
a(n) = Sum_{k=0..n} binomial(n,k)*(1+(-1)^k)/2. - Paul Barry, May 27 2003
a(n) = floor((1+2^n)/2). - Toby Bartels (toby+sloane(AT)math.ucr.edu), Aug 27 2003
G.f.: Sum_{i>=0} x^i/(1-x)^i. - Jon Perry, Jul 10 2004
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(k+1, n-k)*binomial(2*k, k). - Paul Barry, Mar 18 2005
a(n) = Sum_{k=0..floor(n/2)} A055830(n-k, k). - Philippe Deléham, Oct 22 2006
a(n) = Sum_{k=0..n} A098158(n,k). - Philippe Deléham, Dec 04 2006
G.f.: 1/(1 - (x + x^2 + x^3 + ...)). - Geoffrey Critzer, Aug 30 2008
a(n) = A000079(n) - A131577(n).
a(n) = A173921(A000079(n)). - Reinhard Zumkeller, Mar 04 2010
a(n) = Sum_{k=2^n..2^(n+1)-1} A093873(k)/A093875(k), sums of rows of the full tree of Kepler's harmonic fractions. - Reinhard Zumkeller, Oct 17 2010
E.g.f.: (exp(2*x)+1)/2 = (G(0) + 1)/2; G(k) = 1 + 2*x/(2*k+1 - x*(2*k+1)/(x + (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Dec 03 2011
A051049(n) = p(n+1) where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 0, 1, ..., n. - Michael Somos, Apr 18 2012
A008619(n) = p(-1) where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 0, 1, ..., n. - Michael Somos, Apr 18 2012
INVERT transform is A122367. MOBIUS transform is A123707. EULER transform of A059966. PSUM transform is A000079. PSUMSIGN transform is A078008. BINOMIAL transform is A007051. REVERT transform is A105523. A002866(n) = a(n)*n!. - Michael Somos, Apr 18 2012
G.f.: U(0), where U(k) = 1 + x*(k+3) - x*(k+2)/U(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 10 2012
a(n) = A000041(n) + A056823(n). - Omar E. Pol, Aug 31 2013
E.g.f.: E(0), where E(k) = 1 + x/( 2*k+1 - x/E(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 25 2013
G.f.: 1 + x/(1 + x)*( 1 + 3*x/(1 + 3*x)*( 1 + 5*x/(1 + 5*x)*( 1 + 7*x/(1 + 7*x)*( 1 + ... )))). - Peter Bala, May 27 2017
a(n) = Sum_{k=0..2} stirling2(n, k).
G.f.: Sum_{j=0..k} A248925(k,j)*x^j / Product_{j=1..k} 1-j*x with k=2. - Robert A. Russell, Apr 25 2018
a(n) = A053120(n, n), n >= 0, (main diagonal of triangle of Chebyshev's T polynomials). - Wolfdieter Lang, Nov 26 2019

Additional comments from Emeric Deutsch, May 14 2001

Typo corrected by Philippe Deléham, Oct 25 2008

A056272 Word structures of length n using a 5-ary alphabet.

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 202, 855, 3845, 18002, 86472, 422005, 2079475, 10306752, 51263942, 255514355, 1275163905, 6368612302, 31821472612, 159042661905, 795019337135, 3974515030652, 19870830712482, 99348921288655

Offset: 0

Marks R. Nester

Permuting the alphabet will not change a word structure. Thus aabc and bbca have the same structure.
Density of regular language L over {1,2,3,4}^* (i.e., number of strings of length n in L) described by regular expression 11* + 11*2(1+2)* + 11*2(1+2)*3(1+2+3)* + 11*2(1+2)*3(1+2+3)*4(1+2+3+4)* + 11*2(1+2)*3(1+2+3)*4(1+2+3+4)*5(1+2+3+4+5)* - Nelma Moreira, Oct 10 2004
Number of set partitions of [n] into at most 5 parts. - Joerg Arndt, Apr 18 2014

			For a(4)=15, the 7 achiral patterns are AAAA, AABB, ABAB, ABBA, ABBC, ABCA, and ABCD; the 8 chiral patterns are the 4 pairs AAAB-ABBB, AABA-ABAA, AABC-ABCC, and ABAC-ABCB.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Muniru A Asiru, Table of n, a(n) for n = 0..1400 (first 200 terms from Vincenzo Librandi)
Joerg Arndt and N. J. A. Sloane, Counting Words that are in "Standard Order"
Nelma Moreira and Rogerio Reis, On the density of languages representing finite set partitions, Technical Report DCC-2004-07, August 2004, DCC-FC& LIACC, Universidade do Porto
Nelma Moreira and Rogerio Reis, On the Density of Languages Representing Finite Set Partitions, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.8.
Index entries for linear recurrences with constant coefficients, signature (11,-41,61,-30).

Cf. A000351, A007581, A056273, A008290, A007051.
A row of the array in A278984.
Cf. A056324 (unoriented), A320935 (chiral), A305751 (achiral).

List([0..25],n->Sum([0..5],k->Stirling2(n,k))); # Muniru A Asiru, Oct 30 2018

I:=[1,1,2,5,15]; [n le 5 select I[n] else 11*Self(n-1)-41*Self(n-2)+61*Self(n-3)-30*Self(n-4): n in [1..30]]; // Vincenzo Librandi, Apr 19 2014

seq(add(combinat:-stirling2(n, j), j=0..5), n=0..23); # Zerinvary Lajos, Dec 04 2007
# Alternative:
(x*(x*(x*(11*x-37)+32)-10)+1)/(x*(x*(x*(30*x-61)+41)-11)+1):
series(%, x, 32): seq(coeff(%, x, n), n=0..23); # Peter Luschny, Nov 05 2018

CoefficientList[Series[(1 - 10 x + 32 x^2 - 37 x^3 + 11 x^4)/((x - 1) (3 x - 1) (2 x - 1) (5 x - 1)), {x, 0, 30}], x] (* Vincenzo Librandi, Apr 19 2014 *)
LinearRecurrence[{11,-41,61,-30},{1,1,2,5,15},30] (* Harvey P. Dale, Feb 25 2018 *)
Table[Sum[StirlingS2[n, k], {k, 0, 5}], {n, 0, 30}] (* Robert A. Russell, Apr 25 2018 *)
CoefficientList[Series[1/120 (44 + 45 E^x + 20 E^(2 x) + 10 E^(3 x) + E^(5 x)), {x, 0, 30}], x]*Table[k!, {k, 0, 30}] (* Stefano Spezia, Nov 06 2018 *)

a(n) = sum(k=0,5, stirling(n, k, 2) ); \\ Joerg Arndt, Apr 18 2014

a(n) = Sum_{k=0..5} Stirling2(n, k).
a(n) = (5^n + 10*3^n + 20*2^n + 45)/5! for n >= 1. - Vladeta Jovovic, Aug 17 2003
From Nelma Moreira, Oct 10 2004: (Start)
For c=5, a(n) = c^n/c! + Sum_{k=0..c-2} (k^n/k!*(Sum_{j=2..c-k} (-1)^j/j!)).
a(n) = Sum_{k=1..c} g(k, c)*k^n where g(1, 1) = 1, g(1, c) = g(1, c-1) + (-1)^(c-1)/(c-1)! if c > 1; g(k, c) = g(k-1, c-1)/k if c > 1, 2 <= k <= c and n >= 1. (End)
a(n+1) is the top entry of the vector M^n*[1,1,1,1,1,0,0,0,...], where M is an infinite bidiagonal matrix with M(r,r+1)=1 in the superdiagonal and M(r,r)=r, r>=1 as the main diagonal, and the rest zeros. The n-th power of the matrix is multiplied from the right with a column vector starting with 5 1's. - Gary W. Adamson, Jun 24 2011
G.f.: (1 - 10x + 32x^2 - 37x^3 + 11x^4)/((1 - x)*(1 - 2x)*(1 - 3x)*(1 - 5x)). - R. J. Mathar, Jul 06 2011 [Adapted to offset 0 by Robert A. Russell, Oct 30 2018]
G.f.: Sum_{j=0..k} A248925(k,j)*x^j / Product_{j=1..k} 1-j*x with k=5. - Robert A. Russell, Apr 25 2018
E.g.f.: (1/120)*(44 + 45*exp(x) + 20*exp(2*x) + 10*exp(3*x) + exp(5*x)). - Stefano Spezia, Nov 06 2018

a(0)=1 prepended by Robert A. Russell, Nov 06 2018

A124302 Number of set partitions with at most 3 blocks; number of Dyck paths of height at most 4; dimension of space of symmetric polynomials in 3 noncommuting variables.

Original entry on oeis.org

1, 1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842, 29525, 88574, 265721, 797162, 2391485, 7174454, 21523361, 64570082, 193710245, 581130734, 1743392201, 5230176602, 15690529805, 47071589414, 141214768241, 423644304722, 1270932914165, 3812798742494, 11438396227481

Offset: 0

Mike Zabrocki, Oct 25 2006

Row sums of triangle in A056241. - Philippe Deléham, Oct 30 2006
Row sums of triangle in A147746. - Philippe Deléham, Dec 04 2008
Hankel transform is := [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...]. - Philippe Deléham, Dec 04 2008
Number of nonisomorphic graded posets with 0 and 1 and uniform Hasse graph of rank n with no 3-element antichain. (Uniform used in the sense of Retakh, Serconek and Wilson.  Graded used in Stanley's sense that every maximal chain has the same length n.) - David Nacin, Feb 26 2012
Number of Dyck paths of length 2n and height at most 4. - Ira M. Gessel, Aug 06 2012

			There are 15 set partitions of {1,2,3,4}, only {{1},{2},{3},{4}} has more than 3 blocks, so a(4) = 14.
G.f. = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 41*x^5 + 122*x^6 + 365*x^7 + ...

R. Stanley, Enumerative combinatorics, Vol. 1, Cambridge University Press, Cambridge, 1997, pp. 96-100.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Per Alexandersson and Frether Getachew, An involution on derangements, arXiv:2105.08455 [math.CO], 2021.
N. Bergeron, C. Reutenauer, M. Rosas and M. Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, arXiv:math/0502082 [math.CO], 2005; Canad. J. Math. 60 (2008), no. 2, 266-296.
Giulio Cerbai, Anders Claesson, and Luca Ferrari, Stack sorting with restricted stacks, arXiv:1907.08142 [cs.DS], 2019.
S. Felsner and D. Heldt, Lattice Path Enumeration and Toeplitz Matrices, J. Int. Seq. 18 (2015) # 15.1.3.
Daniel Heldt, On the mixing time of the face flip-and up/down Markov chain for some families of graphs, Dissertation, Mathematik und Naturwissenschaften der Technischen Universitat Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften, 2016.
M. Hyatt and J. Remmel, The classification of 231-avoiding permutations by descents and maximum drop, arXiv preprint arXiv:1208.1052 [math.CO], 2012. - From _N. J. A. Sloane_, Dec 24 2012
V. Jelinek, T. Mansour, and M. Shattuck, On multiple pattern avoiding set partitions, Adv. Appl. Math. 50 (2) (2013) 292-326, - From _N. J. A. Sloane_, Jan 01 2013
Sergey Kitaev, Jeffrey Remmel and Mark Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv:1201.6243 [math.CO], 2012-2014 (Corollary 3, case k=4, pages 10-11). - From _N. J. A. Sloane_, May 09 2012
Sergey Kitaev, Jeffrey Remmel, and Mark Tiefenbruck, Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II, Electronic Journal of Combinatorial Number Theory, Volume 15 #A16. (arXiv:1302.2274)
László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
V. Retakh, S. Serconek, and R. Wilson, Hilbert Series of Algebras Associated to Directed Graphs and Order Homology, arXiv:1010.6295 [math.RA], 2010-2011.
Santiago Rojas-Rojas, Camila Muñoz, Edgar Barriga, Pablo Solano, Aldo Delgado, and Carla Hermann-Avigliano, Analytic Evolution for Complex Coupled Tight-Binding Models: Applications to Quantum Light Manipulation, arXiv:2310.12366 [quant-ph], 2023. See p. 12.
M. Rosas and B. Sagan, Symmetric Functions in Noncommuting Variables, Transactions of the American Mathematical Society, 358 (2006), no. 1, 215-232.
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A056272, A123183, A001519, A000110, A008277, A038754, A048328, A068911, A211216.

Essentially the same as A007051.

I:=[1, 1, 2]; [n le 3 select I[n] else  4*Self(n-1) - 3*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Dec 25 2012

a:= proc(n); if n<3 then [1,1,2][n+1]; else 4*a(n-1)-3*a(n-2); fi; end:
# Mike Zabrocki, Oct 25 2006
with(GraphTheory): G:=PathGraph(5): A:= AdjacencyMatrix(G): nmax:=27; for n from 0 to 2*nmax do B(n):=A^n; b(n):=B(n)[1,1]; od: for n from 0 to nmax do a(n):=b(2*n) od: seq(a(n),n=0..nmax);
# Johannes W. Meijer, May 29 2010

a=Exp[x]-1; Range[0, 20]! CoefficientList[Series[1+a+a^2/2+a^3/6, {x,0,20}],x]
Join[{1}, LinearRecurrence[{4, -3}, {1, 2}, 20]] (* David Nacin, Feb 26 2012 *)
CoefficientList[Series[1 / (1 - x / (1 - x / (1 - x / (1 - x)))), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 25 2012 *)
Table[Sum[StirlingS2[n,k],{k,0,3}],{n,0,30}] (* Robert A. Russell, Mar 29 2018 *)

{a(n) = if( n<1, n==0, (3^(n-1) + 1) / 2)}; /* Michael Somos, Apr 03 2014 */

def a(n, adict={0:1, 1:1, 2:2}):
    if n in adict:
        return adict[n]
    adict[n]=4*a(n-1) - 3*a(n-2)
    return adict[n] # David Nacin, Mar 04 2012

O.g.f.: (q^2 - 3*q + 1)/(3*q^2 - 4*q + 1) = Sum_{k=0..3} (q^k/Product_{i=1..k} (1-i*q)).
a(n) = 4*a(n-1) - 3*a(n-2); a(0) = 1, a(1) = 1, a(2) = 2, a(n) = Sum_{k=1..3} A008277(n,k).
Inverse binomial transform of A007581. - Philippe Deléham, Oct 30 2006
a(n) = Sum_{k=0..n} A056241(n,k), n >= 1. - Philippe Deléham, Oct 30 2006
a(0) = 1, a(n) = (3^(n-1) + 1)/2 for n >= 1, see A007051. - Philippe Deléham, Oct 30 2006
E.g.f.: (2 + 3*exp(x) + exp(3x))/6.
G.f.: 1 / (1 - x / (1 - x / (1 - x / (1 - x)))). - Michael Somos, May 03 2012
G.f.: 1 + x + 3*x^2*U(0)/2 where U(k) = 1 + 2/(3*3^k + 3*3^k/(1 - 18*x*3^k/ (9*x*3^k - 1/U(k+1)))); (continued fraction, 4-step). - Sergei N. Gladkovskii, Nov 01 2012
G.f.: 1+x*G(0) where G(k) = 1 + 2*x/( 1-2*x - x*(1-2*x)/(x + (1-2*x)*2/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 10 2012
a(n) = Sum_{k=0..3} Stirling2(n,k). - Robert A. Russell, Mar 29 2018
G.f.: Sum_{j=0..k} A248925(k,j)*x^j / Product_{j=1..k} 1-j*x with k=3. - Robert A. Russell, Apr 25 2018

A056273 Word structures of length n using a 6-ary alphabet.

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 203, 876, 4111, 20648, 109299, 601492, 3403127, 19628064, 114700315, 676207628, 4010090463, 23874362200, 142508723651, 852124263684, 5101098232519, 30560194493456, 183176170057707, 1098318779272060, 6586964947803695, 39510014478620232, 237013033135668883

Offset: 0

Marks R. Nester

Set partitions of the n-set into at most 6 parts; also restricted growth strings (RGS) with six letters s(1),s(2),...,s(6) where the first occurrence of s(j) precedes the first occurrence of s(k) for all j < k. - Joerg Arndt, Jul 06 2011
Permuting the alphabet will not change a word structure. Thus aabc and bbca have the same structure.
Density of regular language L over {1,2,3,4,5,6}^* (i.e., number of strings of length n in L) described by regular expression with c=6: Sum_{i=1..c} Product_{j=1..i} (j(1+...+j)*) where Sum stands for union and Product for concatenation. - Nelma Moreira, Oct 10 2004
Word structures of length n using an N-ary alphabet are generated by taking M^n* the vector [(N 1's),0,0,0,...], leftmost column term = a(n+1). In the case of A056273, the vector = [1,1,1,1,1,1,0,0,0,...]. As the vector approaches all 1's, the leftmost column terms approach A000110, the Bell sequence. - Gary W. Adamson, Jun 23 2011
From Gary W. Adamson, Jul 06 2011: (Start)
Construct an infinite array of sequences representing word structures of length n using an N-ary alphabet as follows:
.
  1, 1, 1,  1,  1,   1,   1,    1, ...; N=1, A000012
  1, 2, 4,  8, 16,  32,  64,  128, ...; N=2, A000079
  1, 2, 5, 14, 41, 122, 365, 1094, ...; N=3, A007051
  1, 2, 5, 15, 51, 187, 715, 2795, ...; N=4, A007581
  1, 2, 5, 15, 52, 202, 855, 3845, ...; N=5, A056272
  1, 2, 5, 15, 52, 203, 876, 4111, ...; N=6, A056273
  ...
The sequences tend to A000110. Finite differences of columns reinterpreted as rows generate A008277 as a triangle: (1; 1,1; 1,3,1; 1,7,6,1; ...). (End)

			For a(4) = 15, the 7 achiral patterns are AAAA, AABB, ABAB, ABBA, ABBC, ABCA, and ABCD; the 8 chiral patterns are the 4 pairs AAAB-ABBB, AABA-ABAA, AABC-ABCC, and ABAC-ABCB.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Muniru A Asiru, Table of n, a(n) for n = 0..1275
Joerg Arndt and N. J. A. Sloane, Counting Words that are in "Standard Order"
Olli-Samuli Lehmus, Optimized Static Allocation of Signal Processing Tasks onto Signal Processing Cores, Master's Thesis, Aalto Univ. (Finland, 2023). See p. 35.
Nelma Moreira and Rogerio Reis, dcc-2004-07.ps
Nelma Moreira and Rogerio Reis, On the density of languages representing finite set partitions, Technical Report DCC-2004-07, August 2004, DCC-FC& LIACC, Universidade do Porto.
Nelma Moreira and Rogerio Reis, On the Density of Languages Representing Finite Set Partitions, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.8.
Index entries for linear recurrences with constant coefficients, signature (16,-95,260,-324,144).

A row of the array in A278984 and A320955.
Cf. A000351, A000400, A007581, A056272, A008290, A007051, A099263, A099262, A000110, A008277.
Cf. A056325 (unoriented), A320936 (chiral), A305752 (chiral).

List([0..25],n->Sum([0..6],k->Stirling2(n,k))); # Muniru A Asiru, Oct 30 2018

[(&+[StirlingSecond(n, i): i in [0..6]]): n in [0..30]]; // Vincenzo Librandi, Nov 07 2018

egf := (265+264*exp(x)+135*exp(x*2)+40*exp(x*3)+15*exp(x*4)+exp(6*x))/6!:
ser := series(egf,x,30): seq(n!*coeff(ser,x,n),n=0..22); # Peter Luschny, Nov 06 2018

Table[Sum[StirlingS2[n,k],{k,0,6}],{n,0,30}] (* or *) LinearRecurrence[ {16,-95,260,-324,144},{1,1,2,5,15,52},30] (* Harvey P. Dale, Jun 05 2015 *)

Vec((1 - 15*x + 81*x^2 - 192*x^3 + 189*x^4 - 53*x^5)/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)*(1-6*x)) + O(x^30)) \\ Michel Marcus, Nov 07 2018

a(n) = Sum_{k=0..6} Stirling2(n, k).
For n > 0, a(n) = (1/6!)*(6^n + 15*4^n + 40*3^n + 135*2^n + 264). - Vladeta Jovovic, Aug 17 2003
From Nelma Moreira, Oct 10 2004: (Start)
For n > 0 and c = 6:
a(n) = (c^n)/c! + Sum_{k=0..c-2} ((k^n)/k!*(Sum_{j=2..c-k}(((-1)^j)/j!))).
a(n) = Sum_{k=1..c} (g(k, c)*k^n) where g(1, 1) = 1; g(1, c) = g(1, c-1) + ((-1)^(c-1))/(c-1)! if c>1. For 2 <= k <= c: g(k, c) = g(k-1, c-1)/k if c>1.  (End)
G.f.: (1 - 15*x + 81*x^2 - 192*x^3 + 189*x^4 - 53*x^5)/((1-x)*(1-2x)*(1-3x)*(1-4x)*(1-6x)). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009 [corrected by R. J. Mathar, Sep 16 2009] [Adapted to offset 0 by Robert A. Russell, Nov 06 2018]
G.f.: Sum_{j=0..k} A248925(k,j)*x^j / Product_{j=1..k} 1-j*x with k=6. - Robert A. Russell, Apr 25 2018
E.g.f.: (265 + 264*exp(x) + 135*exp(x*2) + 40*exp(x*3) + 15*exp(x*4) + exp(6*x))/6!. - Peter Luschny, Nov 06 2018

a(0)=1 prepended by Robert A. Russell, Nov 06 2018

cp's OEIS Frontend

A011782 Coefficients of expansion of (1-x)/(1-2*x) in powers of x.

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A056272 Word structures of length n using a 5-ary alphabet.

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A124302 Number of set partitions with at most 3 blocks; number of Dyck paths of height at most 4; dimension of space of symmetric polynomials in 3 noncommuting variables.

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A056273 Word structures of length n using a 6-ary alphabet.

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A124303 Number of set partitions of length <= 4; sum of first 4 columns of triangle of Stirling numbers of 2nd kind; dimension of space of symmetric polynomials in 4 noncommuting variables.

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A099262 a(n) = (1/5040)7^n + (1/240)5^n + (1/72)4^n + (1/16)3^n + (11/60)*2^n + 53/144. Partial sum of Stirling numbers of second kind S(n,i), i=1..7 (i.e., a(n) = Sum_{i=1..7} S(n,i)).

A099263 a(n) = (1/40320)8^n + (1/1440)6^n + (1/360)5^n + (1/64)4^n + (11/180)3^n + (53/288)2^n + 103/280. Partial sum of Stirling numbers of second kind S(n,i), i=1..8 (i.e., a(n) = Sum_{i=1..8} S(n,i)).