A000969 -id:A000969 - cp's OEIS frontend

A002295 Number of dissections of a polygon: binomial(6n,n)/(5n+1).

1, 1, 6, 51, 506, 5481, 62832, 749398, 9203634, 115607310, 1478314266, 19180049928, 251857119696, 3340843549855, 44700485049720, 602574657427116, 8175951659117794, 111572030260242090, 1530312970340384580, 21085148778264281865, 291705220704719165526

Offset: 0

N. J. A. Sloane

From Wolfdieter Lang, Sep 14 2007: (Start)
a(n), n >= 1, enumerates sextic (6-ary) trees (rooted, ordered, incomplete) with n vertices (including the root).
Pfaff-Fuss-Catalan sequence C^{m}_n for m=6. See the Graham et al. reference, p. 347. eq. 7.66. See also the Pólya-Szegő reference.
Also 6-Raney sequence. See the Graham et al. reference, p. 346-7. (End)
This is instance k = 6 of the generalized Catalan family {C(k, n)}A130564.%20-%20_Wolfdieter%20Lang">{n>=0} given in a comment of A130564. - _Wolfdieter Lang, Feb 05 2024

			There are a(2)=6 sextic trees (vertex degree <= 6 and 6 possible branchings) with 2 vertices (one of them the root). Adding one more branch (one more vertex) to these 6 trees yields 6*6 + binomial(6,2) = 51 = a(3) such trees.

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 23.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, pp. 200, 347.
G. Pólya and G. Szegő, Problems and Theorems in Analysis, Springer-Verlag, Heidelberg, New York, 2 vols., 1972, Vol. 1, problem 211, p. 146 with solution on p. 348.
Ulrike Sattler, Decidable classes of formal power series with nice closure properties, Diplomarbeit im Fach Informatik, Univ. Erlangen - Nürnberg, Jul 27 1994
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).
Editor's note: "Über die Bestimmung der Anzahl der verschiedenen Arten, auf welche sich ein n-Eck durch Diagonalen in lauter m-Ecke zerlegen laesst, mit Bezug auf einige Abhandlungen der Herren Lamé, Rodrigues, Binet, Catalan und Duhamel in dem Journal de Mathématiques pures et appliquées, publié par Joseph Liouville. T. III. IV.", Archiv der Mathematik u. Physik, 1 (1841), pp. 193ff; see especially p. 198.

T. D. Noe, Table of n, a(n) for n = 0..100
V. E. Adler and A. B. Shabat, Volterra chain and Catalan numbers, arXiv:1810.13198 [nlin.SI], 2018.
Paul Barry, On the Gap-sum and Gap-product Sequences of Integer Sequences, arXiv:2104.05593 [math.CO], 2021.
Wun-Seng Chou, Tian-Xiao He, and Peter J.-S. Shiue, On the Primality of the Generalized Fuss-Catalan Numbers, J. Int. Seqs., Vol. 21 (2018), #18.2.1.
Olivier Danvy, Summa Summarum: Moessner's Theorem without Dynamic Programming, arXiv:2412.03127 [cs.DM], 2024. See p. 25.
F. Harary, E. M. Palmer, and R. C. Read, On the cell-growth problem for arbitrary polygons, computer printout, circa 1974
F. Harary, E. M. Palmer, and R. C. Read, On the cell-growth problem for arbitrary polygons, Discr. Math. 11 (1975), 371-389.
Clemens Heuberger, Sarah J. Selkirk, and Stephan Wagner, Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo k, arXiv:2204.14023 [math.CO], 2022.
V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 288
R. P. Loh, A. G. Shannon, and A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, Preprint, 1980.
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014.
Yuan (Friedrich) Qiu, Joe Sawada, and Aaron Williams, Maximize the Rightmost Digit: Gray Codes for Restricted Growth Strings, WALCOM 2025. See p. 5.
B. Sury, Generalized Catalan numbers: linear recursion and divisibility, JIS 12 (2009), Article 09.7.5.
L. Takacs, Enumeration of rooted trees and forests, Math. Scientist 18 (1993), 1-10, esp. Eq. (5).
Jun Yan, Lattice paths enumerations weighted by ascent lengths, arXiv:2501.01152 [math.CO], 2025. See p. 7.

Cf. A001764, A002293, A002294, A002296, A130564.

Fifth column of triangle A062993.

List([0..22],n->Binomial(6*n,n)/(5*n+1)); # Muniru A Asiru, Nov 01 2018

[Binomial(6*n, n)/(5*n + 1): n in [0..20]]; // Vincenzo Librandi, Mar 17 2015

A002295:=n->binomial(6*n, n)/(5*n + 1); seq(A002295(n), n=0..20); # Wesley Ivan Hurt, Jan 29 2014
n:=20:G:=series(RootOf(g = 1+x*g^6, g),x=0,n+1):seq(coeff(G,x,k),k=0..n); # Robert FERREOL, Apr 03 2015

Table[Binomial[6n, n]/(5n + 1), {n, 0, 20}] (* Stefan Steinerberger, Apr 06 2006 *)

A002295(n)=binomial(6*n,n)/(5*n+1) \\ M. F. Hasler, Apr 08 2015

O.g.f.: A(x) = 1 + x*A(x)^6 = 1/(1-x*A(x)^5).
a(n) = binomial(6*n,n-1)/n, n >= 1, a(0)=1. From the Lagrange series of the o.g.f. A(x) with its above given implicit equation.
a(n) = upper left term in M^n, M = the production matrix:
   1,  1
   5,  5,  1
  15, 15,  5,  1
  35, 35, 15,  5,  1
  ...
where (1, 5, 15, 35, ...) = A000332 starting with 1. - Gary W. Adamson, Jul 08 2011
a(n) are special values of Jacobi polynomials, in Maple notation:
  a(n) = JacobiP(n-1, 5*n+1, -n, 1)/n, n=1, 2, ... . - Karol A. Penson, Mar 17 2015
a(n) = binomial(6*n+1, n)/(6*n+1) = A062993(n+4,4). - Robert FERREOL, Apr 03 2015
a(0) = 1; a(n) = Sum_{i1+i2+...+i6=n-1} a(i1)*a(i2)*...*a(i6) for n>=1. - Robert FERREOL, Apr 03 2015
D-finite with recurrence: 5*n*(5*n+1)*(5*n-3)*(5*n-2)*(5*n-1)*a(n) - 72*(6*n-5)*(6*n-1)*(3*n-1)*(2*n-1)*(3*n-2)*a(n-1) = 0. - R. J. Mathar, Sep 06 2016
From Ilya Gutkovskiy, Jan 15 2017: (Start)
O.g.f.: 5F4(1/6,1/3,1/2,2/3,5/6; 2/5,3/5,4/5,6/5; 46656*x/3125).
E.g.f.: 5F5(1/6,1/3,1/2,2/3,5/6; 2/5,3/5,4/5,1,6/5; 46656*x/3125).
a(n) ~ 3^(6*n+1/2)*64^n/(sqrt(Pi)*5^(5*n+3/2)*n^(3/2)). (End)
x*A'(x)/A(x) = (A(x) - 1)/(- 5*A(x) + 6) = x + 11*x^2 + 136*x^3 + 1771*x^4 + ... = (1/6)*Sum_{n >= 1} binomial(6*n,n)*x^n. Cf. A001764 and A002293 - A002296. - Peter Bala, Feb 04 2022
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^11). - Seiichi Manyama, Jun 16 2025

More terms from Stefan Steinerberger, Apr 06 2006

Edited by M. F. Hasler, Apr 08 2015

A001840 Expansion of g.f. x/((1 - x)^2*(1 - x^3)).

Original entry on oeis.org

0, 1, 2, 3, 5, 7, 9, 12, 15, 18, 22, 26, 30, 35, 40, 45, 51, 57, 63, 70, 77, 84, 92, 100, 108, 117, 126, 135, 145, 155, 165, 176, 187, 198, 210, 222, 234, 247, 260, 273, 287, 301, 315, 330, 345, 360, 376, 392, 408, 425, 442, 459, 477, 495, 513, 532, 551, 570, 590

Offset: 0

N. J. A. Sloane

a(n-3) is the number of aperiodic necklaces (Lyndon words) with 3 black beads and n-3 white beads.
Number of triangular partitions (see Almkvist).
Consists of arithmetic progression quadruples of common difference n+1 starting at A045943(n). Refers to the least number of coins needed to be rearranged in order to invert the pattern of a (n+1)-rowed triangular array. For instance, a 5-rowed triangular array requires a minimum of a(4)=5 rearrangements (shown bracketed here) for it to be turned upside down.
  .....{*}..................{*}*.*{*}{*}
  .....*.*....................*.*.*.{*}
  ....*.*.*....---------\......*.*.*
  ..{*}*.*.*...---------/.......*.*
  {*}{*}*.*{*}..................{*}
  - Lekraj Beedassy, Oct 13 2003
Partial sums of 1,1,1,2,2,2,3,3,3,4,4,4,... - Jon Perry, Mar 01 2004
Sum of three successive terms is a triangular number in natural order starting with 3: a(n)+a(n+1)+a(n+2) = T(n+2) = (n+2)*(n+3)/2. - Amarnath Murthy, Apr 25 2004
Apply Riordan array (1/(1-x^3),x) to n. - Paul Barry, Apr 16 2005
Absolute values of numbers that appear in A145919. - Matthew Vandermast, Oct 28 2008
In the Moree definition, (-1)^n*a(n) is the 3rd Witt transform of A033999 and (-1)^n*A004524(n) with 2 leading zeros dropped is the 2nd Witt transform of A033999. - R. J. Mathar, Nov 08 2008
Column sums of:
1 2 3 4 5 6 7  8  9.....
      1 2 3 4  5  6.....
            1  2  3.....
........................
----------------------
1 2 3 5 7 9 12 15 18  - Jon Perry, Nov 16 2010
a(n) is the sum of the positive integers <= n that have the same residue modulo 3 as n. They are the additive counterpart of the triple factorial numbers. - Peter Luschny, Jul 06 2011
a(n+1) is the number of 3-tuples (w,x,y) with all terms in {0,...,n} and w=3*x+y. - Clark Kimberling, Jun 04 2012
a(n+1) is the number of pairs (x,y) with x and y in {0,...,n}, x-y = (1 mod 3), and x+y < n. - Clark Kimberling, Jul 02 2012
a(n+1) is the number of partitions of n into two sorts of part(s) 1 and one sort of (part) 3. - Joerg Arndt, Jun 10 2013
Arrange A004523 in rows successively shifted to the right two spaces and sum the columns:
1  2  2  3  4  4  5  6  6...
      1  2  2  3  4  4  5...
            1  2  2  3  4...
                  1  2  2...
                        1...
------------------------------
1  2  3  5  7  9 12 15 18... - L. Edson Jeffery, Jul 30 2014
a(n) = A258708(n+1,1) for n > 0. - Reinhard Zumkeller, Jun 23 2015
Also the number of triples of positive integers summing to n + 4, the first less than each of the other two. Also the number of triples of positive integers summing to n + 2, the first less than or equal to each of the other two. - Gus Wiseman, Oct 11 2020
Also the lower matching number of the (n+1)-triangular honeycomb king graph = n-triangular grid graph (West convention). - Eric W. Weisstein, Dec 14 2024

			G.f. = x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 9*x^6 + 12*x^7 + 15*x^8 + 18*x^9 + ...
1+2+3=6=t(3), 2+3+5=t(4), 5+7+9=t(5).
[n] a(n)
--------
[1] 1
[2] 2
[3] 3
[4] 1 + 4
[5] 2 + 5
[6] 3 + 6
[7] 1 + 4 + 7
[8] 2 + 5 + 8
[9] 3 + 6 + 9
a(7) = floor(2/3) +floor(3/3) +floor(4/3) +floor(5/3) +floor(6/3) +floor(7/3) +floor(8/3) +floor(9/3) = 12. - _Bruno Berselli_, Aug 29 2013

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 73, problem 25.
Ulrich Faigle, Review of Gerhard Post and G.J. Woeginger, Sports tournaments, home-away assignments and the break minimization problem, MR2224983(2007b:90134), 2007.
Hansraj Gupta, Partitions of j-partite numbers into twelve or a smaller number of parts. Collection of articles dedicated to Professor P. L. Bhatnagar on his sixtieth birthday. Math. Student 40 (1972), 401-441 (1974).
Richard K. Guy, A problem of Zarankiewicz, in P. Erdős and G. Katona, editors, Theory of Graphs (Proceedings of the Colloquium, Tihany, Hungary), Academic Press, NY, 1968, pp. 119-150, (p. 126, divided by 2).
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..1000
Chwas Ahmed, Paul Martin, and Volodymyr Mazorchuk, On the number of principal ideals in d-tonal partition monoids, arXiv preprint arXiv:1503.06718 [math.CO], 2015.
Gert Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., 7 (No. 4, 1998), pp. 343-359.
David J. Broadhurst, On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory, arXiv:hep-th/9604128, 1996.
David Broadhurst and Xavier Roulleau, Number of partitions of modular integers, arXiv:2502.19523 [math.NT], 2025. See p. 19.
Cristian Cobeli, Aaditya Raghavan, and Alexandru Zaharescu, On the central ball in a translation invariant involutive field, arXiv:2408.01864 [math.NT], 2024. See p. 7.
Neville de Mestre and John Baker, Pebbles, Ducks and Other Surprises, Australian Maths. Teacher, Vol. 48, No 3, 1992, pp. 4-7.
Peter M. Chema, Illustration of first 27 terms as corners of a double hexagon spiral from 0
H. Gupta, Partitions of j-partite numbers into twelve or a smaller number of parts, Math. Student 40 (1972), 401-441 (1974). [Annotated scanned copy]
Richard K. Guy, A problem of Zarankiewicz, Research Paper No. 12, Dept. of Math., Univ. Calgary, Jan. 1967. [Annotated and scanned copy, with permission]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 207.
Clark Kimberling, A Combinatorial Classification of Triangle Centers on the Line at Infinity, J. Int. Seq., Vol. 22 (2019), Article 19.5.4.
Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.
R. P. Loh, A. G. Shannon, and A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, Preprint, 1980.
Pieter Moree, The formal series Witt transform, Discr. Math. no. 295 vol. 1-3 (2005) 143-160.
Brian O'Sullivan and Thomas Busch, Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 8a, lambda=3]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Gerhard Post and G. J. Woeginger, Sports tournaments, home-away assignments and the break minimization problem, Discrete Optimization 3, pp. 165-173, 2006.
Michael Somos, Somos Polynomials.
Gary E. Stevens, A Connell-Like Sequence, J. Integer Seqs., 1 (1998), Article 98.1.4.
Andrei K. Svinin, On some class of sums, arXiv:1610.05387 [math.CO], 2016. See p. 7.
Eric Weisstein's World of Mathematics, Lower Matching Number.
Eric Weisstein's World of Mathematics, Triangular Grid Graph.
Eric Weisstein's World of Mathematics, Triangular Honeycomb King Graph.
Index entries for sequences related to Lyndon words.
Index entries for two-way infinite sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A000031, A001037, A008748, A051168.
Ordered union of triangular matchstick numbers A045943 and generalized pentagonal numbers A001318.
Cf. A058937.
A column of triangle A011847.
Cf. A000217, A130205.
Cf. A258708.
A001399 counts 3-part partitions, ranked by A014612.
A337483 counts either weakly increasing or weakly decreasing triples.
A337484 counts neither strictly increasing nor strictly decreasing triples.
A014311 ranks 3-part compositions, with strict case A337453.
Cf. A007052, A007304, A011782, A069905, A156040, A218004.

a001840 n = a001840_list !! n
a001840_list = scanl (+) 0 a008620_list
-- Reinhard Zumkeller, Apr 16 2012

[ n le 2 select n else n*(n+1)/2-Self(n-1)-Self(n-2): n in [1..58] ];  // Klaus Brockhaus, Oct 01 2009

A001840 := n->floor((n+1)*(n+2)/6);
A001840:=-1/((z**2+z+1)*(z-1)**3); # conjectured (correctly) by Simon Plouffe in his 1992 dissertation
seq(floor(binomial(n-1,2)/3), n=3..61); # Zerinvary Lajos, Jan 12 2009
A001840 :=  n -> add(k, k = select(k -> k mod 3 = n mod 3, [$1 .. n])): seq(A001840(n), n = 0 .. 58); # Peter Luschny, Jul 06 2011

a[0]=0; a[1]=1; a[n_]:= a[n]= n(n+1)/2 -a[n-1] -a[n-2]; Table[a[n], {n,0,100}]
f[n_] := Floor[(n + 1)(n + 2)/6]; Array[f, 59, 0] (* Or *)
CoefficientList[ Series[ x/((1 + x + x^2)*(1 - x)^3), {x, 0, 58}], x] (* Robert G. Wilson v *)
a[ n_] := With[{m = If[ n < 0, -3 - n, n]}, SeriesCoefficient[ x /((1 - x^3) (1 - x)^2), {x, 0, m}]]; (* Michael Somos, Jul 11 2011 *)
LinearRecurrence[{2,-1,1,-2,1},{0,1,2,3,5},60] (* Harvey P. Dale, Jul 25 2011 *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n+4,{3}],#[[1]]<#[[2]]&&#[[1]]<#[[3]]&]],{n,0,15}] (* Gus Wiseman, Oct 05 2020 *)

{a(n) = (n+1) * (n+2) \ 6}; /* Michael Somos, Feb 11 2004 */

[binomial(n, 2) // 3 for n in range(2, 61)] # Zerinvary Lajos, Dec 01 2009

a(n) = (A000217(n+1) - A022003(n-1))/3;
a(n) = (A016754(n+1) - A010881(A016754(n+1)))/24;
a(n) = (A033996(n+1) - A010881(A033996(n+1)))/24.
Euler transform of length 3 sequence [2, 0, 1].
a(3*k-1) = k*(3*k + 1)/2;
a(3*k)   = 3*k*(k + 1)/2;
a(3*k+1) = (k + 1)*(3*k + 2)/2.
a(n) = floor( (n+1)*(n+2)/6 ) = floor( A000217(n+1)/3 ).
a(n+1) = a(n) + A008620(n) = A002264(n+3). - Reinhard Zumkeller, Aug 01 2002
From Michael Somos, Feb 11 2004: (Start)
G.f.: x / ((1-x)^2 * (1-x^3)).
a(n) = 1 + a(n-1) + a(n-3) - a(n-4).
a(-3-n) = a(n). (End)
a(n) = a(n-3) + n for n > 2; a(0)=0, a(1)=1, a(2)=2. - Paul Barry, Jul 14 2004
a(n) = binomial(n+3, 3)/(n+3) + cos(2*Pi*(n-1)/3)/9 + sqrt(3)sin(2*Pi*(n-1)/3)/9 - 1/9. - Paul Barry, Jan 01 2005
From Paul Barry, Apr 16 2005: (Start)
a(n) = Sum_{k=0..n} k*(cos(2*Pi*(n-k)/3 + Pi/3)/3 + sqrt(3)*sin(2*Pi*(n-k)/3 + Pi/3)/3 + 1/3).
a(n) = Sum_{k=0..floor(n/3)} n-3*k. (End)
For n > 1, a(n) = A000217(n) - a(n-1) - a(n-2); a(0)=0, a(1)=1.
G.f.: x/(1 + x + x^2)/(1 - x)^3. - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009
a(n) = (4 + 3*n^2 + 9*n)/18 + ((n mod 3) - ((n-1) mod 3))/9. - Klaus Brockhaus, Oct 01 2009
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5), with n>4, a(0)=0, a(1)=1, a(2)=2, a(3)=3, a(4)=5. - Harvey P. Dale, Jul 25 2011
a(n) = A214734(n + 2, 1, 3). - Renzo Benedetti, Aug 27 2012
G.f.: x*G(0), where G(k) = 1 + x*(3*k+4)/(3*k + 2 - 3*x*(k+2)*(3*k+2)/(3*(1+x)*k + 6*x + 4 - x*(3*k+4)*(3*k+5)/(x*(3*k+5) + 3*(k+1)/G(k+1)))); (continued fraction). - Sergei N. Gladkovskii, Jun 10 2013
Empirical: a(n) = floor((n+3)/(e^(6/(n+3))-1)). - Richard R. Forberg, Jul 24 2013
a(n) = Sum_{i=0..n} floor((i+2)/3). - Bruno Berselli, Aug 29 2013
0 = a(n)*(a(n+2) + a(n+3)) + a(n+1)*(-2*a(n+2) - a(n+3) + a(n+4)) + a(n+2)*(a(n+2) - 2*a(n+3) + a(n+4)) for all n in Z. - Michael Somos, Jan 22 2014
a(n) = n/2 + floor(n^2/3 + 2/3)/2. - Bruno Berselli, Jan 23 2017
a(n) + a(n+1) = A000212(n+2). - R. J. Mathar, Jan 14 2021
Sum_{n>=1} 1/a(n) = 20/3 - 2*Pi/sqrt(3). - Amiram Eldar, Sep 27 2022
E.g.f.: (exp(x)*(4 + 12*x + 3*x^2) - 4*exp(-x/2)*cos(sqrt(3)*x/2))/18. - Stefano Spezia, Apr 05 2023

A002296 Number of dissections of a polygon: binomial(7n,n)/(6n+1).

Original entry on oeis.org

1, 1, 7, 70, 819, 10472, 141778, 1997688, 28989675, 430321633, 6503352856, 99726673130, 1547847846090, 24269405074740, 383846168712104, 6116574500860880, 98106248306858715, 1582638261961640247, 25661404527790252375, 417980115131315136400

Offset: 0

N. J. A. Sloane

a(n), n>=1, enumerates heptic (7-ary) trees (rooted, ordered, incomplete) with n vertices (including the root).
Pfaff-Fuss-Catalan sequence C^{m}_n for m=7. See the Graham et al. reference, p. 347. eq. 7.66. See also the Pólya-Szegő reference.
Also 7-Raney sequence. See the Graham et al. reference, pp. 346-347.
a(n) = A258708(3*n,2*n) for n > 0. - Reinhard Zumkeller, Jun 23 2015
This is instance k = 7 of the generalized Catalan family {C(k, n)}A130564.%20-%20_Wolfdieter%20Lang">{n>=0} given in a comment of A130564. - _Wolfdieter Lang, Feb 05 2024

			There are a(2)=7 heptic trees (vertex degree <= 7 and 7 possible branchings) with 2 vertices (one of them the root). Adding one more branch (one more vertex) to these 7 trees yields 7*7 + binomial(7,2) = 70 = a(3) such trees.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, pp. 200, 347.
G. Pólya and G. Szegő, Problems and Theorems in Analysis, Springer-Verlag, Heidelberg, New York, 2 vols., 1972, Vol. 1, problem 211, p. 146 with solution on p. 348.
Ulrike Sattler, Decidable classes of formal power series with nice closure properties, Diplomarbeit im Fach Informatik, Univ. Erlangen - Nürnberg, Jul 27 1994.
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
Wun-Seng Chou, Tian-Xiao He, and Peter J.-S. Shiue, On the Primality of the Generalized Fuss-Catalan Numbers, J. Int. Seqs., Vol. 21 (2018), #18.2.1.
Olivier Danvy, Summa Summarum: Moessner's Theorem without Dynamic Programming, arXiv:2412.03127 [cs.DM], 2024. See p. 25.
F. Harary, E. M. Palmer, and R. C. Read, On the cell-growth problem for arbitrary polygons, computer printout, circa 1974.
F. Harary, E. M. Palmer and R. C. Read, On the cell-growth problem for arbitrary polygons, Discr. Math. 11 (1975), 371-389. See Table 6. n = 8. Sequence U(8) p. 387.
Clemens Heuberger, Sarah J. Selkirk, and Stephan Wagner, Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo k, arXiv:2204.14023 [math.CO], 2022.
V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 289.
R. P. Loh, A. G. Shannon, and A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, Preprint, 1980.
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv:1403.5962 [math.CO], 2014.
Editor's note: Über die Bestimmung der Anzahl der verschiedenen Arten, auf welche sich ein n-Eck durch Diagonalen in lauter m-Ecke zerlegen laesst, mit Bezug auf einige Abhandlungen der Herren Lame, Rodrigues, Binet, Catalan und Duhamel in dem Journal de Mathematiques pures et appliquees, publie par Joseph Liouville. T. III. IV., Archiv der Mathematik u. Physik, 1 (1841), pp. 193ff; see especially p. 198.
B. Sury, Generalized Catalan numbers: linear recursion and divisibility, JIS 12 (2009), Article 09.7.5.
Lajos Takacs, Enumeration of rooted trees and forests, Math. Scientist 18 (1993), 1-10, esp. Eq. (5).
Jun Yan, Lattice paths enumerations weighted by ascent lengths, arXiv:2501.01152 [math.CO], 2025. See p. 7.

Cf. A001764, A002293, A002294, A002295.
Sixth column of triangle A062993.
Cf. A235535: binomial(9n,3n)/(6n+1); A235536: binomial(8n,2n)/(6n+1).
Cf. A258708.
Cf. A130564.

a002296 n = a002296_list !! n
a002296_list = [a258708 (4 * n) (3 * n) | n <- [1..]]
-- Reinhard Zumkeller, Jun 23 2015

seq(binomial(7*n+1, n)/(7*n+1), n=0..30); # Robert FERREOL, Apr 02 2015
n:=30: G:=series(RootOf(g = 1+x*g^7, g), x=0, n+1): seq(coeff(G, x, k), k=0..n); # Robert FERREOL, Apr 02 2015

Table[Binomial[7n,n]/(6n+1),{n,0,20}] (* Harvey P. Dale, Nov 21 2011 *)

a(n)=binomial(7*n,n)/(6*n+1) \\ Charles R Greathouse IV, Feb 06 2012

O.g.f. A(x) = 1 + x*A(x)^7 = 1/(1-x*A(x)^6).
a(n) = binomial(7*n,n-1)/n, n>=1, a(0)=1. From the Lagrange series of the o.g.f. A(x) with its above given implicit equation.
D-finite with recurrence: 72*n*(6*n-1)*(3*n-1)*(2*n-1)*(3*n-2)*(6*n+1)*a(n) - 7*(7*n-3)*(7*n-6)*(7*n-2)*(7*n-5)*(7*n-1)*(7*n-4)*a(n-1) = 0. - R. J. Mathar, Nov 16 2012
a(n) are special values of Jacobi polynomials, in Maple notation:
  a(n) = JacobiP(n-1, 6*n+1, -n, 1)/n, n = 1, 2, ... . - Karol A. Penson, Mar 16 2015
a(n) = binomial(7*n+1, n)/(7*n+1) = A062993(n+5,5). - Robert FERREOL, Apr 02 2015
a(0) = 1; a(n) = Sum_{i1+i2+...+i7=n-1} a(i1)*a(i2)*...*a(i7) for n>=1. - Robert FERREOL, Apr 02 2015
From Ilya Gutkovskiy, Jan 16 2017: (Start)
O.g.f.: 6F5(1/7,2/7,3/7,4/7,5/7,6/7; 1/3,1/2,2/3,5/6,7/6; 823543*x/46656).
E.g.f.: 6F6(1/7,2/7,3/7,4/7,5/7,6/7; 1/3,1/2,2/3,5/6,1,7/6; 823543*x/46656).
a(n) ~ 7^(7*n+1/2)/(sqrt(Pi)*3^(6*n+3/2)*4^(3*n+1)*n^(3/2)). (End)
x*A'(x)/A(x) = (A(x) - 1)/(- 6*A(x) + 7) = x + 13*x^2 + 190*x^3 + 2925*x^4 + ... = (1/7)*Sum_{n >= 1} binomial(7*n,n)*x^n. Cf. A001764 and A002293, A002294, A002295. - Peter Bala, Feb 04 2022
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^13). - Seiichi Manyama, Jun 16 2025

Pfaff-Fuss-Catalan, Raney, o.g.f. and 7-ary tree comments from Wolfdieter Lang, Sep 14 2007

A004773 Numbers congruent to {0, 1, 2} mod 4: a(n) = floor(4*n/3).

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 8, 9, 10, 12, 13, 14, 16, 17, 18, 20, 21, 22, 24, 25, 26, 28, 29, 30, 32, 33, 34, 36, 37, 38, 40, 41, 42, 44, 45, 46, 48, 49, 50, 52, 53, 54, 56, 57, 58, 60, 61, 62, 64, 65, 66, 68, 69, 70, 72, 73, 74, 76, 77, 78, 80, 81, 82, 84, 85, 86, 88, 89, 90

Offset: 0

N. J. A. Sloane

The sequence b(n) = floor((4/3)*(n+2)) appears as an upper bound in Fijavz and Wood.
Binary expansion does not end in 11.
From Guenther Schrack, May 04 2023: (Start)
The sequence is the interleaving of the sequences A008586, A016813, A016825, in that order.
Let S(n) = a(n) + a(n+1) + a(n+2). Then floor(S(n)/3) = A042968(n+1), round(S(n)/3) = a(n+1), ceiling(S(n)/3) = A042965(n+2). (End)

Daniel Starodubtsev, Table of n, a(n) for n = 0..10000
Gasper Fijavz and David R. Wood, Graph Minors and Minimum Degree, arXiv:0812.1064 [math.CO], 2008.
Niall Graham and Frank Harary, Edge Sums of Hypercubes, Bull. Irish Math. Soc., Vol. 21 (1988), pp. 8-12.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A177702 (first differences), A000969 (partial sums).
Cf. A032766, this sequence, A001068, A047226, A047368, A004777.
Cf. similar sequences with formula n+i*floor(n/3) listed in A281899.
Cf. A002264, A004396, A004523, A004767, A004772, A008586, A010872, A016813, A016825, A042965, A042968.

[n: n in [0..100] | n mod 4 in [0..2]]; // Vincenzo Librandi, Dec 23 2010

seq(floor(n/3)+n,n=0..68); # Gary Detlefs, Mar 20 2010

f[n_] := Floor[4 n/3]; Array[f, 69, 0] (* Robert G. Wilson v, Dec 24 2010 *)
fQ[n_] := Mod[n, 4] != 3; Select[ Range[0, 90], fQ] (* Robert G. Wilson v, Dec 24 2010 *)
a[0] = 0; a[n_] := a[n] = a[n - 1] + 2 - If[ Mod[a[n - 1], 4] < 2, 1, 0]; Array[a, 69, 0] (* Robert G. Wilson v, Dec 24 2010 *)
CoefficientList[ Series[x (1 + x + 2 x^2)/((1 - x) (1 - x^3)), {x, 0, 68}], x] (* Robert G. Wilson v, Dec 24 2010 *)

a(n)=4*n\3 \\ Charles R Greathouse IV, Sep 27 2012

G.f.: x*(1+x+2*x^2)/((1-x)*(1-x^3)).
a(0) = 0, a(n+1) = a(n) + a(n) mod 4 + 0^(a(n) mod 4). - Reinhard Zumkeller, Mar 23 2003
a(n) = A004396(n) + A004523(n); complement of A004767. - Reinhard Zumkeller, Aug 29 2005
a(n) = floor(n/3) + n. - Gary Detlefs, Mar 20 2010
a(n) = (12*n-3+3*cos(2*n*Pi/3)+sqrt(3)*sin(2*n*Pi/3))/9. - Wesley Ivan Hurt, Sep 30 2017
E.g.f.: (3*exp(x)*(4*x - 1) + exp(-x/2)*(3*cos((sqrt(3)*x)/2) + sqrt(3)*sin((sqrt(3)*x)/2)))/9. - Stefano Spezia, Jun 09 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)-1)*Pi/8 + sqrt(2)*log(sqrt(2)+2)/4 + (2-sqrt(2))*log(2)/8. - Amiram Eldar, Dec 05 2021
From Guenther Schrack, May 04 2023: (Start)
a(n) = (12*n - 3 +  w^(2*n)*(w + 2) - w^n*(w - 1))/9 where w = (-1 + sqrt(-3))/2.
a(n) = 2*floor(n/3) + floor((n+1)/3) + floor((n+2)/3).
a(n) = (4*n - n mod 3)/3.
a(n) = a(n-3) + 4.
a(n) = a(n-1) + a(n-3) - a(n-4).
a(n) = 4*A002264(n) + A010872(n).
a(n) = A042968(n+1) - 1.
(End)

A258708 Triangle read by rows: T(i,j) = integer part of binomial(i+j, i-j)/(2*j+1) for i >= 1 and j = 0..i-1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 7, 4, 1, 1, 7, 14, 12, 5, 1, 1, 9, 25, 30, 18, 6, 1, 1, 12, 42, 66, 55, 26, 7, 1, 1, 15, 66, 132, 143, 91, 35, 8, 1, 1, 18, 99, 245, 334, 273, 140, 45, 9, 1, 1, 22, 143, 429, 715, 728, 476, 204, 57, 10, 1

Offset: 1

N. J. A. Sloane, Jun 12 2015

In the Loh-Shannon-Horadam paper, Table 3 contains a typo (see Extensions lines).
T(n,k) = round(A258993(n,k)/(2*k+1)). - Reinhard Zumkeller, Jun 22 2015
From Reinhard Zumkeller, Jun 23 2015: (Start)
(using tables 4 and 5 of the Loh-Shannon-Horadam paper, p. 8f).
T(n, n-1)  = 1;
T(n, n-2)  = n for n > 1;
T(n, n-3)  = A000969(n-3) for n > 2;
T(n, n-4)  = A000330(n-3) for n > 3;
T(n, n-5)  = T(2*n-7, 2) = A000970(n) for n > 4;
T(n, n-6)  = A000971(n) for n > 5;
T(n, n-7)  = A000972(n) for n > 6;
T(n, n-8)  = A000973(n) for n > 7;
T(n, 1)    = A001840(n-1) for n > 1;
T(2*n, n)  = A001764(n);
T(3*n-1, 1) = A000326(n);
T(3*n, 2*n) = A002294(n);
T(4*n, 3*n) = A002296(n). (End)

			Triangle T(i, j) (with rows i >= 1 and columns j >= 0) begins as follows:
  1;
  1,  1;
  1,  2,  1;
  1,  3,  3,   1;
  1,  5,  7,   4,   1;
  1,  7, 14,  12,   5,   1;
  1,  9, 25,  30,  18,   6,   1;
  1, 12, 42,  66,  55,  26,   7,  1;
  1, 15, 66, 132, 143,  91,  35,  8, 1;
  1, 18, 99, 245, 334, 273, 140, 45, 9, 1;
  ...

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
R. P. Loh, A. G. Shannon, and A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, Preprint, 1980.

Cf. A011793.
Cf. A007318, A258993, A158405, A005408, A006013 (central terms).
Cf. A000326, A000330, A000969, A000970, A000971, A000972, A000973, A000976, A001764, A001840, A002294, A002296.

a258708 n k = a258708_tabl !! (n-1) !! k
a258708_row n = a258708_tabl !! (n-1)
a258708_tabl = zipWith (zipWith ((round .) . ((/) `on` fromIntegral)))
                       a258993_tabl a158405_tabl
-- Reinhard Zumkeller, Jun 22 2015, Jun 16 2015

Corrected T(8,5) = 26 from Reinhard Zumkeller, Jun 13 2015

A092498 Expansion of g.f. (1 + x + 2x^2)/((1 - x)^3(1 - x^3)).

Original entry on oeis.org

1, 4, 11, 23, 41, 67, 102, 147, 204, 274, 358, 458, 575, 710, 865, 1041, 1239, 1461, 1708, 1981, 2282, 2612, 2972, 3364, 3789, 4248, 4743, 5275, 5845, 6455, 7106, 7799, 8536, 9318, 10146, 11022, 11947, 12922, 13949, 15029, 16163, 17353, 18600, 19905, 21270

Offset: 0

N. J. A. Sloane, Apr 05 2004

Arises from the Molien series for 4-dimensional group of structure S_3 X C_2 and order 12, which preserves the complete weight enumerators of even trace-Hermitian self-dual additive codes over GF(4). The Molien series is (1 + x^2 + 2*x^4)/((1 - x^2)^3 *(1 - x^6)).
From Philippe LALLOUET (philip.lallouet(AT)wanadoo.fr), Sep 24 2007: (Start)
Also arises when a pyramid is built row by row with squares of size 1.
At the first step, we put a single square on row 1. At the second, we put a square above the first one, on row 2, and a square on each of its sides on row 1. At each following step, we begin a new row with one square and add a square at each end of each of the previous rows. The term a(n) of the sequence is the total number of squares of any size which can be seen in the entire triangular array.
..........................
.....................|__|..
.....|__|.....|__||__|
||..||__||...||__||__||
The table below gives the number of squares by size, and the total number of squares (i.e., a(n)), for each row.
  +-----------------------+
  |size size size size    |
n |  1    2    3    4     | a(n)
--+-----------------------+-----
1 | .1....................|....1
2 | .4....................|....4
3 | .9....2...............|...11
4 | 16....6....1..........|...23
5 | 25...12....4..........|...41
6 | 36...20....9....2.....|...67
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
G. Nebe, E. M. Rains, and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-3,3,-1).

Cf. A014126.

Cf. A000969 (first differences). - R. J. Mathar, Jan 05 2009

A092498:=n->(4*n^3+21*n^2+35*n+18-6*floor((n+2)/3))/18; seq(A092498(n), n=0..50); # Wesley Ivan Hurt, Apr 19 2014

Table[(4*n^3 + 21*n^2 + 35*n + 18 - 6*Floor[(n + 2)/3])/18, {n, 0, 50}] (* Wesley Ivan Hurt, Apr 19 2014 *)
CoefficientList[Series[(1 + x + 2 x^2)/((1 - x)^3 (1 -x^3)), {x, 0, 40 }], x] (* Vincenzo Librandi, Apr 20 2014 *)
LinearRecurrence[{3,-3,2,-3,3,-1},{1,4,11,23,41,67},50] (* Harvey P. Dale, Jul 08 2017 *)

a(n) = (4*n^3+21*n^2+35*n+18-6*floor((n+2)/3))/18. - Luce ETIENNE, Apr 18 2014
a(n) = Sum_{j=0..floor(2*n/3)} ((4*n+5-6*j-(-1)^j)/4)*((4*n+3-6*j+(-1)^j)/4). - Luce ETIENNE, Oct 28 2014
E.g.f.: exp(-x/2)*(3*exp(3*x/2)*(2 + x)*(8 + 25*x + 4*x^2) + 6*cos(sqrt(3)*x/2) - 2*sqrt(3)*sin(sqrt(3)*x/2))/54. - Stefano Spezia, Apr 05 2023

Edited by N. J. A. Sloane, May 15 2014

A004210 "Magic" integers: a(n+1) is the smallest integer m such that there is no overlap between the sets {m, m-a(i), m+a(i): 1 <= i <= n} and {a(i), a(i)-a(j), a(i)+a(j): 1 <= j < i <= n}.

Original entry on oeis.org

1, 3, 8, 18, 30, 43, 67, 90, 122, 161, 202, 260, 305, 388, 416, 450, 555, 624, 730, 750, 983, 1059, 1159, 1330, 1528, 1645, 1774, 1921, 2140, 2289, 2580, 2632, 2881, 3158, 3304, 3510, 3745, 4086, 4563, 4741, 4928, 5052, 5407, 5864, 6242, 6528, 6739, 7253

Offset: 1

N. J. A. Sloane, following a suggestion from B. G. DeBoer, Dec 15 1978

The definition implies that the sets {a(i)} (A004210), {a(i)-a(j), j < i} (A206522) and {a(i)+a(j), j < i} (A206523) are disjoint. A206524 gives the complement of their union.

R. A. Bates, E. Riccomagno, R. Schwabe, H. P. Wynn, Lattices and dual lattices in optimal experimental design for Fourier models, Computational Statistics & Data Analysis Volume 28, Issue 3, 4 September 1998, Pages 283-296. See page 293.
D. R. Hofstadter, "Goedel, Escher, Bach: An Eternal Golden Braid", Basic Books Incorporated, p. 73
P. Mark Kayll, Well-spread sequences and edge-labelings with constant Hamiltonian weight, Disc. Math. & Theor. Comp. Sci 6 2 (2004) 401-408
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 = 1..150
B. G. DeBoer, Letter to N. J. A. Sloane, Dec 15 1978, with enclosure of Silvertom article.
J. V. Silverton, On the generation of 'magic integrals', Acta Cryst. A34 (1978) p. 634.
Eric Weisstein's World of Mathematics, Magic Integer.

Cf. A000969, A005228, A206522, A206523, A206524.

import Data.List (intersect)
a004210 n = a004210_list !! (n-1)
a004210_list = magics 1 [0] [0] where
   magics :: Integer -> [Integer] -> [Integer] -> [Integer]
   magics n ms tests
      | tests `intersect` nMinus == [] && tests `intersect` nPlus == []
      = n : magics (n+1) (n:ms) (nMinus ++ nPlus ++ tests)
      | otherwise
      = magics (n+1) ms tests
      where nMinus = map (n -) ms
            nPlus  = map (n +) ms
-- magics works also as generator for a126428_list, cf. A126428.
-- Reinhard Zumkeller, Mar 03 2011

a[1] = 1; a[n_] := a[n] = Module[{pairs = Flatten[ Table[{a[j] + a[k], a[k] - a[j]}, {j, 1, n-1}, {k, j+1, n-1}]], an = a[n-1] + 1}, While[ True, If[ Intersection[ Join[ Array[a, n-1], pairs], Prepend[ Flatten[ Table[{a[j] + an, an - a[j]}, {j, 1, n-1}]], an]] == {}, Break[], an++]]; an]; Table[a[n], {n, 1, 48}] (* Jean-François Alcover, Nov 10 2011 *)

a(n+1) = min{ k | k and k +- a(i) are not equal to a(i) or a(i)-a(j) or a(i)+a(j) for any n+1 > i > j > 0}. [Corrected by T. D. Noe, Sep 08 2008]

Additional comments from Robert M. Burton, Jr. (bob(AT)oregonstate.edu), Feb 20 2005
More terms from Joshua Zucker, May 04 2006
Edited by N. J. A. Sloane, Sep 06 2008 at the suggestion of R. J. Mathar
Edited by N. J. A. Sloane, Feb 08 2012

A000973 Fermat coefficients.

Original entry on oeis.org

1, 15, 99, 429, 1430, 3978, 9690, 21318, 43263, 82225, 148005, 254475, 420732, 672452, 1043460, 1577532, 2330445, 3372291, 4790071, 6690585, 9203634, 12485550, 16723070, 22137570, 28989675, 37584261, 48275865, 61474519

Offset: 8

N. J. A. Sloane

a(n) = A258708(n,n-8). - Reinhard Zumkeller, Jun 23 2015

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).

Vincenzo Librandi, Table of n, a(n) for n = 8..1000
R. P. Loh, A. G. Shannon, A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, Preprint, 1980.
P. A. Piza, Fermat coefficients, Math. Mag., 27 (1954), 141-146.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (8, -28, 56, -70, 56, -28, 8, -1).

Cf. A053129.

Cf. A258708.

a000973 n = a258708 n (n - 8)  -- Reinhard Zumkeller, Jun 23 2015

[Binomial(2*n-8, 7)/8: n in [8..40]]; // Vincenzo Librandi, Apr 10 2012

A000973:=(z+1)*(z**2+6*z+1)/(z-1)**8; # conjectured by Simon Plouffe in his 1992 dissertation
A000973:=n->binomial(2*n-8, 7)/8; seq(A000973(n), n=8..40); # Wesley Ivan Hurt, Apr 16 2014

CoefficientList[Series[(1+7*x+7*x^2+x^3)/(1-x)^8,{x,0,40}],x] (* Vincenzo Librandi, Apr 10 2012 *)

a(n) = binomial(2*n-8, 7)/8.
G.f.: (x^8)*(1+7*x+7*x^2+x^3)/(1-x)^8.
G.f.: A(x)= (1+7*x+7*x^2+x^3)/(x-1)^8 = 1 + 45*x/(G(0)-45*x), |x|<1; if |x|>1, G(0)=45*x;
G(k) = (k+1)*(2*k+3) + x*(k+5)*(2*k+9) - x*(k+1)*(k+6)*(2*k+3)*(2*k+11)/G(k+1); (continued fraction Euler's 1st kind, 1-step ). - Sergei N. Gladkovskii, Jun 15 2012

More terms from David W. Wilson, Oct 11 2000

A000967 Sum of Fermat coefficients.

Original entry on oeis.org

1, 2, 4, 8, 18, 40, 91, 210, 492, 1165, 2786, 6710, 16267, 39650, 97108, 238824, 589521, 1459960, 3626213, 9030450, 22542396, 56393792, 141358274, 354975429, 892893120, 2249412290, 5674891000, 14335757256, 36259245522, 91815545800

Offset: 1

N. J. A. Sloane

nonn

			n...Sum_{c=1..n} (n:c).....a(n)
--------------------------------
.1........1.................1
.2........2.................2
.3........4.................4
.4........8+1/3.............8

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).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000 (corrected by Sean A. Irvine, April 17, 2019)
R. P. Loh, A. G. Shannon, A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, Preprint, 1980.
P. A. Piza, Fermat coefficients, Math. Mag., 27 (1954), 141-146.

Cf. A143858, A258708.

import Data.Function (on)
a000967 n = round $ sum $
            zipWith ((/) `on` fromIntegral) (a258993_row n) [1, 3 ..]
-- Reinhard Zumkeller, Jun 22 2015

[Round((&+[Binomial(n+k,n-k)/(2*k+1): k in [0..n-1]])): n in [1..35]]; // G. C. Greubel, Apr 16 2019

FermatCoeff:=(n,c)->binomial(2*n-c,c-1)/c:seq(round(add(FermatCoeff(n,c),c=1..n)),n=1..40); # Pab Ter, Oct 13 2005

Table[Round[Sum[Binomial[n+k, n-k]/(2*k+1), {k, 0, n-1}]], {n,1,35}] (* G. C. Greubel, Apr 16 2019 *)

{a(n) = round(sum(k=0,n-1, binomial(n+k,n-k)/(2*k+1)))}; \\ G. C. Greubel, Apr 16 2019

[round(sum(binomial(n+k,n-k)/(2*k+1) for k in (0..n-1))) for n in (1..35)] # G. C. Greubel, Apr 16 2019

Following Piza's definition for the Fermat coefficients: (n:c)=binomial(2n-c, c-1)/c, a(n)= Round( sum_ {c=1..n} (n:c) ). - Pab Ter (pabrlos2(AT)yahoo.com), Oct 13 2005

a(n) = rounded(sum(A258993(n,k)/(2*k+1)): k = 0..n-1). - Reinhard Zumkeller, Jun 22 2015

More terms from Pab Ter (pabrlos2(AT)yahoo.com), Oct 13 2005

cp's OEIS Frontend

A002295 Number of dissections of a polygon: binomial(6n,n)/(5n+1).

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A001840 Expansion of g.f. x/((1 - x)^2*(1 - x^3)).

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A002296 Number of dissections of a polygon: binomial(7n,n)/(6n+1).

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Haskell

Maple

Mathematica

PARI

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Extensions

A004773 Numbers congruent to {0, 1, 2} mod 4: a(n) = floor(4*n/3).

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Magma

Maple

Mathematica

PARI

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A258708 Triangle read by rows: T(i,j) = integer part of binomial(i+j, i-j)/(2*j+1) for i >= 1 and j = 0..i-1.

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Haskell

Extensions

A092498 Expansion of g.f. (1 + x + 2*x^2)/((1 - x)^3*(1 - x^3)).

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A092498 Expansion of g.f. (1 + x + 2x^2)/((1 - x)^3(1 - x^3)).