A008277 -id:A008277 - cp's OEIS frontend

A132056 Triangle read by rows, the Bell transform of Product_{k=0..n} 7*k+1 without column 0.

1, 8, 1, 120, 24, 1, 2640, 672, 48, 1, 76560, 22800, 2160, 80, 1, 2756160, 920160, 104880, 5280, 120, 1, 118514880, 43243200, 5639760, 347760, 10920, 168, 1, 5925744000, 2323918080, 336510720, 24071040, 937440, 20160, 224, 1

Offset: 1

Wolfdieter Lang Sep 14 2007

Previous name was: Triangle of numbers related to triangle A132057; generalization of Stirling numbers of second kind A008277, Lah-numbers A008297, ...
a(n,m) enumerates unordered n-vertex m-forests composed of m plane increasing 8-ary trees. See the F. Bergeron et al. reference, especially Table 1, first row, for the e.g.f. for m=1.
a(n,m) := S2(8; n,m) is the eighth triangle of numbers in the sequence S2(k;n,m), k=1..7: A008277 (unsigned Stirling 2nd kind), A008297 (unsigned Lah), A035342, A035469, A049029, A049385, A092082, respectively. a(n,1)=A045754(n), n>=1.

			{1}; {8,1}; {120,24,1}; {2640,672,48,1}; ...

F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1992, pp. 24-48.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
M. Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3
W. Lang, First 10 rows.

Cf. A132060 (row sums), A132061 (alternating row sums).

Cf. A092082 S2(7) triangle.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> mul(7*k+1, k=0..n), 8); # Peter Luschny, Jan 27 2016

a[n_, m_] := a[n, m] = ((m*a[n-1, m-1]*(m-1)! + (m+7*n-7)*a[n-1, m]*m!)*n!)/(n*m!*(n-1)!);
a[n_, m_] /; n < m = 0; a[_, 0] = 0; a[1, 1] = 1;
Flatten[Table[a[n, m], {n, 1, 8}, {m, 1, n}]][[1 ;; 36]]
(* Jean-François Alcover, Jun 17 2011 *)
rows = 8;
a[n_, m_] := BellY[n, m, Table[Product[7k+1, {k, 0, j}], {j, 0, rows}]];
Table[a[n, m], {n, 1, rows}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018 *)

a(n, m) = n!*A132057(n, m)/(m!*7^(n-m)); a(n+1, m) = (7*n+m)*a(n, m)+ a(n, m-1), n >= m >= 1; a(n, m) := 0, n

E.g.f. of m-th column: ((-1+(1-7*x)^(-1/7))^m)/m!.

a(n, m) = sum(|A051186(n, j)|*S2(j, m), j=m..n) (matrix product), with S2(j, m):= (j, m) (Stirling2 triangle). Priv. comm. with W. Lang by E. Neuwirth, Feb 15 2001; see also the 2001 Neuwirth reference. See the general comment on products of Jabotinsky matrices given under A035342.

New name from Peter Luschny, Jan 27 2016

A002657 Numerators of Cauchy numbers of second type (= Bernoulli numbers B_n^{(n)}).

Original entry on oeis.org

1, 1, 5, 9, 251, 475, 19087, 36799, 1070017, 2082753, 134211265, 262747265, 703604254357, 1382741929621, 8164168737599, 5362709743125, 8092989203533249, 15980174332775873, 12600467236042756559

Offset: 0

N. J. A. Sloane

These coefficients (with alternating signs) are also known as the Nørlund [or Norlund, Noerlund or Nörlund] numbers. [After the Danish mathematician Niels Erik Nørlund (1885-1981). - Amiram Eldar, Jun 17 2021]
The denominators are found in A002790. The alternating rational sequence ((-1)^n)*a(n)/A002790(n)is the z-sequence for the Stirling2 triangle A008277(n+1,k+1), n>=k>=0. This is the Sheffer (exp(x),exp(x)-1) triangle. See the W. Lang link under A006232 for Sheffer a- and z-sequences with references, and the conversion to S. Roman's notation. The a-sequence is A006232(n)/A006233(n). - Wolfdieter Lang, Oct 06 2011 [This is the Sheffer triangle A007318*A048993. Added Jun 20 2017]
A simple series with the signless Cauchy numbers of second type C2(n) leads to Euler's constant: gamma = 1 - Sum_{n >=1} C2(n)/(n*(n+1)!) = 1 - 1/4 - 5/72 - 1/32 - 251/14400 - 19/1728 - 19087/2540160 - ..., see references [Blagouchine] below, as well as A075266 and A262235. - Iaroslav V. Blagouchine, Sep 15 2015

			1, 1/2, 5/6, 9/4, 251/30, 475/12, 19087/84, 36799/24, 1070017/90, ...

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 294.
P. Curtz, Intégration numérique des systèmes différentiels à conditions initiales, Centre de Calcul Scientifique de l'Armement, Arcueil, 1969.
Louis Melville Milne-Thompson, Calculus of Finite Differences, 1951, p. 136.
N. E. Nørlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, Berlin, 1924.
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).

T. D. Noe, Table of n, a(n) for n = 0..100
Ibrahim M. Alabdulmohsin, The Language of Finite Differences, in Summability Calculus: A Comprehensive Theory of Fractional Finite Sums, Springer, Cham, 2018, pp. 133-149.
Iaroslav V. Blagouchine, Expansions of generalized Euler's constants into the series of polynomials in 1/pi^2 and into the formal enveloping series with rational coefficients only. Journal of Number Theory (Elsevier), vol. 158, pp. 365-396, 2016. arXiv version, arXiv:1501.00740 [math.NT], 2015.
Iaroslav V. Blagouchine, Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to 1/pi, Journal of Mathematical Analysis and Applications (Elsevier), 2016. arXiv version, arXiv:1408.3902 [math.NT], 2014-2016.
Iaroslav V. Blagouchine, Three notes on Ser's and Hasse's representation for the zeta-functions, Integers (2018) 18A, Article #A3.
Donghyun Kim and Jaeseong Oh, Extending the science fiction and the Loehr--Warrington formula, arXiv:2409.01041 [math.CO], 2024. See p. 32.
Takao Komatsu, Convolution Identities for Cauchy Numbers of the Second Kind, Kyushu Journal of Mathematics, Vol. 69, No. 1 (2015), pp. 125-144.
Guodong Liu, Some computational formulas for Norlund numbers, Fib. Quart., Vol. 45, No. 2 (2007), pp. 133-137.
Guo-Dong Liu, H. M. Srivastava, and Hai-Quing Wang, Some Formulas for a Family of Numbers Analogous to the Higher-Order Bernoulli Numbers, J. Int. Seq., Vol. 17 (2014), Article 14.4.6.
Rui-Li Liu and Feng-Zhen Zhao, Log-concavity of two sequences related to Cauchy numbers of two kinds, Online Journal of Analytic Combinatorics, Issue 14 (2019), #09.
Donatella Merlini, Renzo Sprugnoli and M. Cecilia Verri, The Cauchy numbers, Discrete Math., Vol. 306, No. 16 (2006), pp. 1906-1920.
Louis Melville Milne-Thompson, Calculus of Finite Differences, 1951. [Annotated scan of pages 135, 136 only]
N. E. Nørlund, Vorlesungen ueber Differenzenrechnung Springer, 1924, p. 461.
N. E. Nörlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, Berlin, 1924; page 461 [Annotated scanned copy of pages 144-151 and 456-463]
Michael O. Rubinstein, Identities for the Riemann zeta function, Ramanujan J., Vol. 27, No. 1 (2012), pp. 29-42; arXiv preprint, arXiv:0812.2592 [math.NT], 2008-2009.
Feng-Zhen Zhao, Sums of products of Cauchy numbers, Discrete Math., Vol. 309, No. 12 (2009), pp. 3830-3842.
Index entries for sequences related to Bernoulli numbers.

Cf. A002206, A002207, A002208, A002209, A002790, A006232, A006233, A075266, A075267, A262235.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(-x/((1-x)*Log(1-x)) )); [Numerator(Factorial(n-1)*b[n]): n in [1..m-1]]; // G. C. Greubel, Oct 29 2018

seq(numer(add((-1)^(n-k)*Stirling1(n,k)/(k+1),k=0..n)),n=0..10); # Peter Luschny, Apr 28 2009

Table[Abs[Numerator[NorlundB[n,n]]],{n,0,30}](* Vladimir Joseph Stephan Orlovsky, Dec 30 2010 *)
a[ n_] := If[ n < 0, 0, (-1)^n Numerator @ NorlundB[ n, n]]; (* Michael Somos, Jul 12 2014 *)
a[ n_] := If[ n < 0, 0, Numerator@Integrate[ Pochhammer[ x, n], {x, 0, 1}]]; (* Michael Somos, Jul 12 2014 *)
a[ n_] := If[ n < 0, 0, Numerator[ n! SeriesCoefficient[ -x / ((1 - x) Log[ 1 - x]), {x, 0, n}]]]; (* Michael Somos, Jul 12 2014 *)
a[ n_] := If[ n < 0, 0, (-1)^n Numerator[ n! SeriesCoefficient[ (x / (Exp[x] - 1))^n, {x, 0, n}]]]; (* Michael Somos, Jul 12 2014 *)

v(n):=if n=0 then 1 else 1-sum(v(i)/(n-i+1),i,0,n-1);
makelist(num(n!*v(n)),n,0,10); /* Vladimir Kruchinin, Aug 28 2013 */

Numerator of integral of x(x+1)...(x+n-1) from 0 to 1.
E.g.f.: -x/((1-x)*log(1-x)). (Note: the numerator of the coefficient of x^n/n! is a(n). - Michael Somos, Jul 12 2014). E.g.f. rewritten by Iaroslav V. Blagouchine, May 07 2016
Numerator of Sum_{k=0..n} (-1)^(n-k) A008275(n,k)/(k+1). - Peter Luschny, Apr 28 2009
a(n) = numerator(n!*v(n)), where v(n) = 1 - Sum_{i=0..n-1} v(i)/(n-i+1), v(0)=1. - Vladimir Kruchinin, Aug 28 2013

A001998 Bending a piece of wire of length n+1; walks of length n+1 on a tetrahedron; also non-branched catafusenes with n+2 condensed hexagons.

Original entry on oeis.org

1, 2, 4, 10, 25, 70, 196, 574, 1681, 5002, 14884, 44530, 133225, 399310, 1196836, 3589414, 10764961, 32291602, 96864964, 290585050, 871725625, 2615147350, 7845353476, 23535971854, 70607649841, 211822683802, 635467254244, 1906400965570, 5719200505225, 17157599124190

Offset: 0

N. J. A. Sloane

The wire stays in the plane, there are n bends, each is R,L or O; turning the wire over does not count as a new figure.
Equivalently, walks of n+1 steps on a tetrahedron, visiting n+2 vertices, with n "corners"; the symmetry group is S4, reversing a walk does not count as different. Simply interpret R,L,O as instructions to turn R, turn L, or retrace the last step. Walks are not self-avoiding.
Also, it appears that a(n) gives the number of equivalence classes of n-tuples of 0, 1 and 2, where two n-tuples are equivalent if one can be obtained from the other by a sequence of operations R and C, where R denotes reversal and C denotes taking the 2's complement (C(x)=2-x). This has been verified up to a(19)=290585050. Example: for n=3 there are ten equivalence classes {000, 222}, {001, 100, 122, 221}, {002, 022, 200, 220}, {010, 212}, {011, 110, 112, 211}, {012, 210}, {020, 202}, {021, 102, 120, 201}, {101, 121}, {111}, so a(3)=10. - John W. Layman, Oct 13 2009
There exists a bijection between chains of n+2 hexagons and the above described equivalence classes of n-tuples of 0,1, and 2. Namely, for a given chain of n+2 hexagons we take the sequence of the numbers of vertices of degree 2 (0, 1, or 2) between the consecutive contact vertices on one side of the chain; switching to the other side we obtain the 2's complement of this sequence; reversing the order of the hexagons, we obtain the reverse sequence. The inverse mapping is straightforward. For example, to a linear chain of 7 hexagons there corresponds the 5-tuple 11111. - Emeric Deutsch, Apr 22 2013
If we treat two wire bends (or walks, or tuples) related by turning over (or reversing) as different in any of the above-given interpretations of this sequence, we get A007051 (or A124302). Also, a(n-1) is the sum of first 3 terms in n-th row of A284949, see crossrefs therein. - Andrey Zabolotskiy, Sep 29 2017
a(n-1) is the number of color patterns (set partitions) in an unoriented row of length n using 3 or fewer colors (subsets). - Robert A. Russell, Oct 28 2018
From Allan Bickle, Jun 02 2022: (Start)
a(n) is the number of (unlabeled) 3-paths with n+6 vertices. (A 3-path with order n at least 5 can be constructed from a 4-clique by iteratively adding a new 3-leaf (vertex of degree 3) adjacent to an existing 3-clique containing an existing 3-leaf.)
Recurrences appear in the papers by Bickle, Eckhoff, and Markenzon et al. (End)
a(n) is also the number of distinct planar embeddings of the (n+1)-alkane graph (up to at least n=9, and likely for all n). - Eric W. Weisstein, May 21 2024

			There are 2 ways to bend a piece of wire of length 2 (bend it or not).
For n=4 and a(n-1)=10, the 6 achiral patterns are AAAA, AABB, ABAB, ABBA, ABCA, and ABBC.  The 4 chiral pairs are AAAB-ABBB, AABA-ABAA, AABC-ABCC, and ABAC-ABCB. - _Robert A. Russell_, Oct 28 2018

A. T. Balaban, Enumeration of Cyclic Graphs, pp. 63-105 of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976; see p. 75.
S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, Enumeration of tree-like octagonal systems: catapolyoctagons, ACH Models in Chem. 134 (1997), 55-70.
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.]
R. C. Read, The Enumeration of Acyclic Chemical Compounds, pp. 25-61 of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976. [I think this reference does not mention this sequence. - N. J. A. Sloane, Aug 10 2006]
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).

Indranil Ghosh, Table of n, a(n) for n = 0..2092 (terms 0..500 from T. D. Noe)
A. T. Balaban, J. Brunvoll, B. N. Cyvin, and S. J. Cyvin, Enumeration of branched catacondensed benzenoid hydrocarbons and their numbers of Kekulé structures, Tetrahedron 44(1) (1988), 221-228. See Eq. (6), p. 223.
A. T. Balaban and F. Harary, Chemical graphs V: enumeration and proposed nomenclature of benzenoid cata-condensed polycyclic aromatic hydrocarbons, Tetrahedron 24 (1968), 2505-2516.
Christian Barrientos and Sarah Minion, On the Graceful Cartesian Product of Alpha-Trees, Theory and Applications of Graphs, Vol. 4: Iss. 1, Article 3, 2017. (It mentions this sequence on p. 7.)
L. W. Beineke and R. E. Pippert, On the enumeration of planar trees of hexagons, Glasgow Math. J., 15 (1974), 131-147.
L. W. Beineke and R. E. Pippert, On the enumeration of planar trees of hexagons, Glasgow Math. J., 15 (1974), 131-147 [annotated scanned copy].
Allan Bickle, How to Count k-Paths, J. Integer Sequences, 25 (2022) Article 22.5.6.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, Isomer enumeration of some polygonal systems representing polycyclic conjugated hydrocarbons, Journal of Molecular structure 376 (Issues 1-3) (1996), 495-505. See Table 2 on p. 501.
S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, Unbranched catacondensed polygonal systems containing hexagons and tetragons, Croatica Chem. Acta, 69 (1996), 757-774.
J. Eckhoff, Extremal interval graphs, J. Graph Theory 17 1 (1993), 117-127.
R. M. Foster, Solution to Problem E185, Amer. Math. Monthly, 44 (1937), 50-51.
R. M. Foster, Solution to Problem E185, Amer. Math. Monthly, 44 (1937), 50-51 [annotated scanned copy].
F. Harary and R. W. Robinson, Tapeworms, Unpublished manuscript, circa 1973. (Annotated scanned copy)
Thomas M. Liggett and Wenpin Tang, One-dependent hard-core processes and colorings of the star graph, arXiv:1804.06877 [math.PR], 2018.
L. Markenzon, O. Vernet, and P. R. da Costa Pereira, A clique-difference encoding scheme for labelled k-path graphs, Discrete Appl. Math. 156 (2008), 3216-3222.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Gyula Tasi and Fujio Mizukami, Quantum algebraic-combinatoric study of the conformational properties of n-alkanes, J. Math. Chemistry, 25, 1999, 55-64 (see p. 60).
Eric Weisstein's World of Mathematics, Alkane Graph.
Eric Weisstein's World of Mathematics, Planar Embedding.
Index entries for sequences obtained by enumerating foldings
Index entries for linear recurrences with constant coefficients, signature (4,0,-12,9).

Cf. A036359, A002216, A005963, A000228, A001997, A001444, A038766, A007051.
Column 3 of A320750, offset by one. Column k = 0 of A323942, offset by two.
Cf. A124302 (oriented), A107767 (chiral), A182522 (achiral), with varying offsets.
Column 3 of A320750.
The numbers of unlabeled k-paths for k = 2..7 are given in A005418, A001998, A056323, A056324, A056325, and A345207, respectively.
The sequences above converge to A103293(n+1).

a:=[];; for n in [2..45] do if n mod 2 =0 then Add(a,((3^((n-2)/2)+1)/2)^2); else Add(a,  3^((n-3)/2)+(1/4)*(3^(n-2)+1)); fi; od; a; # Muniru A Asiru, Oct 28 2018

A001998 := proc(n) if n = 0 then 1 elif n mod 2 = 1 then (1/4)*(3^n+4*3^((n-1)/2)+1) else (1/4)*(3^n+2*3^(n/2)+1); fi; end;
A001998:=(-1+3*z+2*z**2-8*z**3+3*z**4)/(z-1)/(3*z-1)/(3*z**2-1); # conjectured by Simon Plouffe in his 1992 dissertation; gives sequence with an extra leading 1

a[n_?OddQ] := (1/4)*(3^n + 4*3^((n - 1)/2) + 1); a[n_?EvenQ] := (1/4)*(3^n + 2*3^(n/2) + 1); Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Jan 25 2013, from formula *)
LinearRecurrence[{4,0,-12,9},{1,2,4,10},30] (* Harvey P. Dale, Apr 10 2013 *)
Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2,k] + Ach[n-2,k-1] + Ach[n-2,k-2]] (* A304972 *)
k=3; Table[Sum[StirlingS2[n,j]+Ach[n,j],{j,k}]/2,{n,40}] (* Robert A. Russell, Oct 28 2018 *)

Vec((1-2*x-4*x^2+6*x^3)/((1-x)*(1-3*x)*(1-3*x^2)) + O(x^50)) \\ Colin Barker, May 15 2016

a(n) = if n mod 2 = 0 then ((3^((n-2)/2)+1)/2)^2 else 3^((n-3)/2)+(1/4)*(3^(n-2)+1).
G.f.: (1-2*x-4*x^2+6*x^3) / ((1-x)*(1-3*x)*(1-3*x^2)). - Corrected by Colin Barker, May 15 2016
a(n) = 4*a(n-1)-12*a(n-3)+9*a(n-4), with a(0)=1, a(1)=2, a(2)=4, a(3)=10. - Harvey P. Dale, Apr 10 2013
a(n) = (1+3^n+3^(1/2*(-1+n))*(2-2*(-1)^n+sqrt(3)+(-1)^n*sqrt(3)))/4. - Colin Barker, May 15 2016
E.g.f.: (2*sqrt(3)*sinh(sqrt(3)*x) + 3*exp(2*x)*cosh(x) + 3*cosh(sqrt(3)*x))/6. - Ilya Gutkovskiy, May 15 2016
From Robert A. Russell, Oct 28 2018: (Start)
a(n-1) = (A124302(n) + A182522(n)) / 2 = A124302(n) - A107767(n-1) = A107767(n-1) + A182522(n).
a(n-1) = Sum_{j=1..k} (S2(n,j) + Ach(n,j)) / 2, where k=3 is the maximum number of colors, S2 is the Stirling subset number A008277, and Ach(n,k) = [n>=0 & n<2 & n==k] + [n>1]*(k*Ach(n-2,k) + Ach(n-2,k-1) + Ach(n-2,k-2)).
a(n-1) = A057427(n) + A056326(n) + A056327(n). (End)
a(2*n) = A007051(n)^2; a(2*n+1) = A007051(n)*A007051(n+1). - Todd Simpson, Mar 25 2024

Offset and Maple code corrected by Colin Mallows, Nov 12 1999

Term added by Robert A. Russell, Oct 30 2018

A223532 Triangle S(n,k) by rows: coefficients of 6^(n/2)(x^(5/6)d/dx)^n when n=0,2,4,6,...

Original entry on oeis.org

1, 1, 6, 7, 84, 36, 91, 1638, 1404, 216, 1729, 41496, 53352, 16416, 1296, 43225, 1296750, 2223000, 1026000, 162000, 7776, 1339975, 48239100, 103369500, 63612000, 15066000, 1446336, 46656, 49579075, 2082321150, 5354540100, 4118877000, 1300698000, 187300512

Offset: 1

Udita Katugampola, Mar 23 2013

			Triangle begins:
1;
1, 6;
7, 84, 36;
91, 1638, 1404, 216;
1729, 41496, 53352, 16416, 1296;
43225, 1296750, 2223000, 1026000, 162000, 7776;
1339975, 48239100, 103369500, 63612000, 15066000, 1446336, 46656;
49579075, 2082321150, 5354540100, 4118877000, 1300698000, 187300512, 12083904, 279936;

U. N. Katugampola, Mellin Transforms of Generalized Fractional Integrals and Derivatives, Appl. Math. Comput. 257(2015) 566-580.
U. N. Katugampola, Existence and Uniqueness results for a class of Generalized Fractional Differential Equations, arXiv preprint arXiv:1411.5229, 2014

Even rows of A223172.

Cf. A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522, A223168-A223172, A223523-A223532.

a[0]:= f(x):
for i from 1 to 20 do
a[i] := simplify(6^((i+1)mod 2)*x^((4((i+1)mod 2)+1)/6)*(diff(a[i-1],x$1 )));
end do:
for j from 1 to 10 do
b[j]:=a[2j];
end do;

A052551 Expansion of 1/((1 - x)(1 - 2x^2)).

Original entry on oeis.org

1, 1, 3, 3, 7, 7, 15, 15, 31, 31, 63, 63, 127, 127, 255, 255, 511, 511, 1023, 1023, 2047, 2047, 4095, 4095, 8191, 8191, 16383, 16383, 32767, 32767, 65535, 65535, 131071, 131071, 262143, 262143, 524287, 524287, 1048575, 1048575, 2097151, 2097151

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Equals row sums of triangle A137865. - Gary W. Adamson, Feb 18 2008
Also, the decimal representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 566", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. - Robert Price, Jul 05 2017
Number of nonempty subsets of {1,2,...,n+1} that contain only odd numbers. a(0) = a(1) = 1: {1}; a(6) = a(7) = 15: {1}, {3}, {5}, {7}, {1,3}, {1,5}, {1,7}, {3,5}, {3,7}, {5,7}, {1,3,5}, {1,3,7}, {1,5,7}, {3,5,7}, {1,3,5,7}. - Enrique Navarrete, Mar 16 2018
Number of nonempty subsets of {1,2,...,n+2} that contain only even numbers. a(0) = a(1) = 1: {2}; a(4) = a(5) = 7: {2}, {4}, {6}, {2,4}, {2,6}, {4,6}, {2,4,6}. - Enrique Navarrete, Mar 26 2018
Doubling of A000225(n+1), n >= 0 entries. First differences give A077957. - Wolfdieter Lang, Apr 08 2018
a(n-2) is the number of achiral rows or cycles of length n partitioned into two sets or the number of color patterns using exactly 2 colors. An achiral row or cycle is equivalent to its reverse. Two color patterns are equivalent if the colors are permuted. For n = 4, the a(n-2) = 3 row patterns are AABB, ABAB, and ABBA; the cycle patterns are AAAB, AABB, and ABAB. For n = 5, the a(n-2) = 3 patterns for both rows and cycles are AABAA, ABABA, and ABBBA. For n = 6, the a(n-2) = 7 patterns for rows are AAABBB, AABABB, AABBAA, ABAABA, ABABAB, ABBAAB, and ABBBBA; the cycle patterns are AAAAAB, AAAABB, AAABAB, AAABBB, AABAAB, AABABB, and ABABAB. - Robert A. Russell, Oct 15 2018
For integers m > 1, the expansion of 1/((1 - x)*(1 - m*x^2)) generates a(n) = (sqrt(m)^(n + 1)*((-1)^n*(sqrt(m) - 1) + sqrt(m) + 1) - 2)/(2*(m - 1)). It appears, for integer values of n >= 0 and m > 1, that it could be simplified in the integral domain a(n) = (m^(1 + floor(n/2)) - 1)/(m - 1). - Federico Provvedi, Nov 23 2018
From Werner Schulte, Mar 04 2019: (Start)
More generally: For some fixed integers q and r > 0 the expansion of A(q,r; x) = 1/((1-x)*(1-q*x^r)) generates coefficients a(q,r; n) = (q^(1+floor(n/r))-1)/(q-1) for n >= 0; the special case q = 1 leads to a(1,r; n) = 1 + floor(n/r).
The a(q,r; n) satisfy for n > r a linear recurrence equation with constant coefficients. The signature vector is given by the sum of two vectors v and w where v has terms 1 followed by r zeros, i.e., (1,0,0,...,0), and w has r-1 leading zeros followed by q and -q, i.e., (0,0,...,0,q,-q).
Let a_i(q,r; n) be the convolution inverse of a(q,r; n). The terms are given by the sum a_i(q,r; n) = b(n) + c(n) for n >= 0 where b(n) has terms 1 and -1 followed by infinitely zeros, i.e., (1,-1,0,0,0,...), and c(n) has r leading zeros followed by -q, q and infinitely zeros, i.e., (0,0,...,0,-q,q,0,0,0,...).
Here is an example for q = 3 and r = 5: The expansion of A(3,5; x) = 1/((1-x)*(1-3*x^5)) = Sum_{n>=0} a(3,5; n)*x^n generates the sequence of coefficients (a(3,5; n)) = (1,1,1,1,1,4,4,4,4,4,13,13,13,13,13,40,...) where r = 5 controls the repetition and q = 3 the different values.
The a(3,5; n) satisfy for n > 5 the linear recurrence equation with constant coefficients and signature (1,0,0,0,0,0) + (0,0,0,0,3,-3) = (1,0,0,0,3,-3).
The convolution inverse a_i(3,5; n) has terms (1,-1,0,0,0,0,0,0,0,...) + (0,0,0,0,0,-3,3,0,0,...) = (1,-1,0,0,0,-3,3,0,0,...).
For further examples and informations see A014983 (q,r = -3,1), A077925 (q,r = -2,1), A000035 (q,r = -1,1), A000012 (q,r = 0,1), A000027 (q,r = 1,1), A000225 (q,r = 2,1), A003462 (q,r = 3,1), A077953 (q,r = -2,2), A133872 (q,r = -1,2), A004526 (q,r = 1,2), A052551 (this sequence with q,r = 2,2), A077886 (q,r = -2,3), A088911 (q,r = -1,3), A002264 (q,r = 1,3) and A077885 (q,r = 2,3). The offsets might be different.
(End)
a(n) is the number of palindromes of length n over the alphabet {1,2} containing the letter 1. More generally, the number of palindromes of length n over the alphabet {1,2,...,k} containing the letter 1 is given by k^ceiling(n/2)-(k-1)^ceiling(n/2). - Sela Fried, Dec 10 2024

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 488
Donatella Merlini and Massimo Nocentini, Algebraic Generating Functions for Languages Avoiding Riordan Patterns, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.3.
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Wolfram Research, Wolfram Atlas of Simple Programs
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (1,2,-2).

Cf. A000225, A077957, A136865, A289404, A289405, A032085, A050686.
Column 2 (offset by two) of A304972.
Cf. A000225 (oriented), A056326 (unoriented), and A122746(n-2) (chiral) for rows.
Cf. A056295 (oriented), A056357 (unoriented), and A059053 (chiral) for cycles.

Flat(List([1..21],n->[2^n-1,2^n-1])); # Muniru A Asiru, Oct 16 2018

[2^Floor(n/2)-1: n in [2..50]]; // Vincenzo Librandi, Aug 16 2011

spec := [S,{S=Prod(Sequence(Prod(Z,Union(Z,Z))),Sequence(Z))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

Table[StirlingS2[Floor[n/2] + 2, 2], {n, 0, 50}] (* Robert A. Russell, Dec 20 2017 *)
Drop[LinearRecurrence[{1, 2, -2}, {0, 1, 1}, 50], 1] (* Robert A. Russell, Oct 14 2018 *)
CoefficientList[Series[1/((1-x)*(1-2*x^2)), {x, 0, 50}], x] (* Stefano Spezia, Oct 16 2018 *)
2^(1+Floor[(Range[0,50])/2])-1 (* Federico Provvedi, Nov 22 2018 *)
((-1)^#(Sqrt[2]-1)+Sqrt[2]+1)2^((#-1)/2)-1&@Range[0, 50] (* Federico Provvedi, Nov 23 2018 *)

x='x+O('x^50); Vec(1/((1-x)*(1-2*x^2))) \\ Altug Alkan, Mar 19 2018

[2^(floor(n/2)) -1 for n in (2..50)] # G. C. Greubel, Mar 04 2019

G.f.: 1/((1 - x)*(1 - 2*x^2)).
Recurrence: a(1) = 1, a(0) = 1, -2*a(n) - 1 + a(n+2) = 0.
a(n) = -1 + Sum((1/2)*(1 + 2*alpha)*alpha^(-1 - n)) where the sum is over alpha = the two roots of -1 + 2*x^2.
a(n) = A016116(n+2) - 1. - R. J. Mathar, Jun 15 2009
a(n) = A060546(n+1) - 1. - Filip Zaludek, Dec 10 2016
From Robert A. Russell, Oct 15 2018: (Start)
a(n-2) = S2(floor(n/2)+1,2), where S2 is the Stirling subset number A008277.
a(n-2) = 2*A056326(n) - A000225(n) = A000225(n) - 2*A122746(n-2) = A056326(n) - A122746(n-2).
a(n-2) = 2*A056357(n) - A056295(n) = A056295(n) - 2*A059053(n) = A056357(n) - A059053(n). (End)
From Federico Provvedi, Nov 22 2018: (Start)
a(n) = 2^( 1 + floor(n/2) ) - 1.
a(n) = ( (-1)^n*(sqrt(2)-1) + sqrt(2) + 1 ) * 2^( (n - 1)/2 ) - 1.  (End)
E.g.f.: 2*cosh(sqrt(2)*x) + sqrt(2)*sinh(sqrt(2)*x) - cosh(x) - sinh(x). - Franck Maminirina Ramaharo, Nov 23 2018

More terms from James Sellers, Jun 06 2000

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

A346426 Number A(n,k) of partitions of the (n+k)-multiset {0,...,0,1,2,...,k} with n 0's; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 5, 5, 4, 3, 15, 15, 11, 7, 5, 52, 52, 36, 21, 12, 7, 203, 203, 135, 74, 38, 19, 11, 877, 877, 566, 296, 141, 64, 30, 15, 4140, 4140, 2610, 1315, 592, 250, 105, 45, 22, 21147, 21147, 13082, 6393, 2752, 1098, 426, 165, 67, 30, 115975, 115975, 70631, 33645, 13960, 5317, 1940, 696, 254, 97, 42

Offset: 0

Alois P. Heinz, Jul 16 2021

Also number A(n,k) of factorizations of 2^n * Product_{i=1..k} prime(i+1); A(3,1) = 7: 2*2*2*3, 2*3*4, 4*6, 2*2*6, 3*8, 2*12, 24; A(1,2) = 5: 2*3*5, 5*6, 3*10, 2*15, 30.

			A(2,2) = 11: 00|1|2, 001|2, 1|002, 0|0|1|2, 0|01|2, 0|1|02, 01|02, 00|12, 0|0|12, 0|012, 0012.
Square array A(n,k) begins:
   1,  1,   2,    5,   15,    52,    203,     877,    4140, ...
   1,  2,   5,   15,   52,   203,    877,    4140,   21147, ...
   2,  4,  11,   36,  135,   566,   2610,   13082,   70631, ...
   3,  7,  21,   74,  296,  1315,   6393,   33645,  190085, ...
   5, 12,  38,  141,  592,  2752,  13960,   76464,  448603, ...
   7, 19,  64,  250, 1098,  5317,  28009,  158926,  963913, ...
  11, 30, 105,  426, 1940,  9722,  52902,  309546, 1933171, ...
  15, 45, 165,  696, 3281, 16972,  95129,  572402, 3670878, ...
  22, 67, 254, 1106, 5372, 28582, 164528, 1015356, 6670707, ...
  ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-10 give: A000041, A000070, A082775, A093802, A346857, A346858, A346859, A346860, A346861, A346862, A346863.
Rows n=0+1, 2-10 give: A000110, A035098, A169587, A169588, A346851, A346852, A346853, A346854, A346855, A346856.
Main diagonal gives A346424.
Antidiagonal sums give A346428.
Cf. A000079, A001055, A008277, A048993, A070826, A126442, A129306, A219727, A255903, A292508, A346520.

s:= proc(n) option remember; expand(`if`(n=0, 1,
      x*add(s(n-j)*binomial(n-1, j-1), j=1..n)))
    end:
S:= proc(n, k) option remember; coeff(s(n), x, k) end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=0,
      combinat[numbpart](n), add(b(n-j, i-1), j=0..n)))
    end:
A:= (n, k)-> add(S(k, j)*b(n, j), j=0..k):
seq(seq(A(n, d-n), n=0..d), d=0..10);

s[n_] := s[n] = Expand[If[n == 0, 1, x Sum[s[n - j] Binomial[n - 1, j - 1], {j, 1, n}]]];
S[n_, k_] := S[n, k] = Coefficient[s[n], x, k];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i == 0, PartitionsP[n], Sum[b[n - j, i - 1], {j, 0, n}]]];
A[n_, k_] := Sum[S[k, j] b[n, j], {j, 0, k}];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Aug 18 2021, after Alois P. Heinz *)

A(n,k) = A001055(A000079(n)*A070826(k+1)).
A(n,k) = Sum_{j=0..k} A048993(k,j)*A292508(n,j+1).
A(n,k) = Sum_{j=0..k} Stirling2(k,j)*Sum_{i=0..n} binomial(j+i-1,i)*A000041(n-i).

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A223172 Triangle S(n,k) by rows: coefficients of 6^((n-1)/2)*(x^(1/6)*d/dx)^n when n is odd, and of 6^(n/2)*(x^(5/6)*d/dx)^n when n is even.

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A134278 A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(6).

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A318762 Number of permutations of a multiset whose multiplicities are the prime indices of n.

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A132056 Triangle read by rows, the Bell transform of Product_{k=0..n} 7*k+1 without column 0.

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A002657 Numerators of Cauchy numbers of second type (= Bernoulli numbers B_n^{(n)}).

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Magma

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A001998 Bending a piece of wire of length n+1; walks of length n+1 on a tetrahedron; also non-branched catafusenes with n+2 condensed hexagons.

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A223532 Triangle S(n,k) by rows: coefficients of 6^(n/2)*(x^(5/6)*d/dx)^n when n=0,2,4,6,...

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A223172 Triangle S(n,k) by rows: coefficients of 6^((n-1)/2)(x^(1/6)d/dx)^n when n is odd, and of 6^(n/2)(x^(5/6)d/dx)^n when n is even.

A223532 Triangle S(n,k) by rows: coefficients of 6^(n/2)(x^(5/6)d/dx)^n when n=0,2,4,6,...

A052551 Expansion of 1/((1 - x)(1 - 2x^2)).