A026815 -id:A026815 - cp's OEIS frontend

A008284 Triangle of partition numbers: T(n,k) = number of partitions of n in which the greatest part is k, 1 <= k <= n. Also number of partitions of n into k positive parts, 1 <= k <= n.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 3, 3, 2, 1, 1, 1, 3, 4, 3, 2, 1, 1, 1, 4, 5, 5, 3, 2, 1, 1, 1, 4, 7, 6, 5, 3, 2, 1, 1, 1, 5, 8, 9, 7, 5, 3, 2, 1, 1, 1, 5, 10, 11, 10, 7, 5, 3, 2, 1, 1, 1, 6, 12, 15, 13, 11, 7, 5, 3, 2, 1, 1, 1, 6, 14, 18, 18, 14, 11, 7, 5, 3, 2, 1, 1, 1, 7, 16, 23, 23, 20, 15, 11, 7, 5, 3, 2, 1, 1

Offset: 1

N. J. A. Sloane

From Frederik Beaujean (beaujean(AT)mpp.mpg.de), Apr 09 2010: (Start)
A000041(n+1) = 1 + Sum_{r=1..n} Sum_{k=1..min(r,n-r+1)} T(r,k).
T(n, n-k) is also the number of partitions of k in which the greatest part is at most n-k. (End)
From Richard R. Forberg, Dec 26 2014: (Start)
Elements of T(n, k) for n <= 2+3k, equal A000041(n-k) - A000070(n-2k-1), with the assumption A000070(n) = 0 for n < 0.
The diagonal T(2+2k, k), for k > 1 equals A007042, and the diagonal T(3+3k,k), for k >= 1, equals A104384. (End)
T(-n, k) is used as a definition for A380038, which can therefore be seen as an extension of this sequence for negative n. - Friedjof Tellkamp, Jan 18 2025

			The triangle T(n,k) begins:
   n\k 1  2  3  4  5  6  7  8  9 10 11 12 ...
   1:  1
   2:  1  1
   3:  1  1  1
   4:  1  2  1  1
   5:  1  2  2  1  1
   6:  1  3  3  2  1  1
   7:  1  3  4  3  2  1  1
   8:  1  4  5  5  3  2  1  1
   9:  1  4  7  6  5  3  2  1  1
  10:  1  5  8  9  7  5  3  2  1  1
  11:  1  5 10 11 10  7  5  3  2  1  1
  12:  1  6 12 15 13 11  7  5  3  2  1  1
... Reformatted and extended by _Wolfdieter Lang_, Dec 03 2012; additional extension by _Bob Selcoe_, Jun 09 2013
T(7,3) = 4 because we have [3,3,1], [3,2,2], [3,2,1,1] and [3,1,1,1,1], each having greatest part 3; or [5,1,1], [4,2,1], [3,3,1] and [3,2,2] each having 3 parts.
* Example from formula above: T(10,4) = 9 because T(6,4) + T(6,3) + T(6,2) + T(6,1) = 2 + 3 + 3 + 1 = 9.
* P(n) = P(n-1) + DT(n-1). P(n) = unordered partitions of n. (A000041) DT(n-1) = sum of diagonals beginning at T(n-1,1).
Example P(11) = 56, P(10) = 42, sum DT(10) = 1 + 4 + 5 + 3 + 1 = 14. - _Bob Selcoe_, Jun 09 2013
From _Omar E. Pol_, Nov 19 2019: (Start)
Illustration of initial terms: T(n,k) is also the number of vertical line segments in the k-th column of the n-th diagram, which represents the partitions of n:
.
    1    1 1    1 1 1    1 2 1 1    1 2 2 1 1    1 3 3 2 1 1    1 3 4 3 2 1 1
.
   _|   _| |   _| | |   _| | | |   _| | | | |   _| | | | | |   _| | | | | | |
        _ _|   _ _| |   _ _| | |   _ _| | | |   _ _| | | | |   _ _| | | | | |
               _ _ _|   _ _ _| |   _ _ _| | |   _ _ _| | | |   _ _ _| | | | |
                        _ _|   |   _ _|   | |   _ _|   | | |   _ _|   | | | |
                        _ _ _ _|   _ _ _ _| |   _ _ _ _| | |   _ _ _ _| | | |
                                   _ _ _|   |   _ _ _|   | |   _ _ _|   | | |
                                   _ _ _ _ _|   _ _ _ _ _| |   _ _ _ _ _| | |
                                                _ _|   |   |   _ _|   |   | |
                                                _ _ _ _|   |   _ _ _ _|   | |
                                                _ _ _|     |   _ _ _|     | |
                                                _ _ _ _ _ _|   _ _ _ _ _ _| |
                                                               _ _ _|   |   |
                                                               _ _ _ _ _|   |
                                                               _ _ _ _|     |
                                                               _ _ _ _ _ _ _|
(End)

L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 94, 96 and 307.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 219.
D. E. Knuth, The Art of Computer Programming, Volume 4, Fascicle 3: Generating All Combinations and Partitions, Addison-Wesley Professional, 2005, pp. 38, 45, 50 [From Frederik Beaujean (beaujean(AT)mpp.mpg.de), Apr 09 2010]
D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.4, p. 400.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIV.2, p. 493.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 294.

Franklin T. Adams-Watters, First 200 rows, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, p. 831. [scanned copy]
Henry Bottomley, Illustration of initial terms
D. J. Broadhurst and D. Kreimer, Towards cohomology of renormalization: bigrading the combinatorial Hopf algebra of rooted trees, arXiv:hep-th/0001202, 2000.
FindStat - Combinatorial Statistic Finder, The length of the partition
Martin Griffiths, Generating Functions for Extended Stirling Numbers of the First Kind, Journal of Integer Sequences, 17 (2014), #14.6.4.
Roser Homs and Anna-Lena Winz, Deformations of local Artin rings via Hilbert-Burch matrices, arXiv:2309.06871 [math.AC], 2023. See p. 16.
Wolfdieter Lang, First 10 rows and more.
T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
OEIS Wiki, Sorting numbers
Tilman Piesk, Illustration of initial terms
Stephan Scholz and Lothar Berger, Fast computation of function composition derivatives for flatness-based control of diffusion problems, J. Math. Industry (2024) Vol. 14, Art. No. 15.
Teerapat Srichan, Watcharapon Pimsert, and Vichian Laohakosol, New Recursion Formulas for the Partition Function, J. Int. Seq., Vol. 24 (2021), Article 21.6.6.
Eric Weisstein's World of Mathematics, Partition Function P
Index entries for triangles generated by the Multiset Transformation

A000041 is row sums and diagonal.
Columns k=1-10 are: A057427, A004526, A069905, A026810, A026811, A026812, A026813, A026814, A026815, A026816.
Partial sums of rows gives A026820.
Cf. A038497, A038498, A039805-A039809, A060016, A126988, A380038.
Read from right to left gives A058398.
Subtriangle of A072233 without row n=0 and column m=0.
Cf. A007042, A104384 which are diagonals with slope -2, -3.

a008284 n k = a008284_tabl !! (n-1) !! (k-1)
a008284_row n = a008284_tabl !! (n-1)
a008284_tabl = [1] : f [[1]] where
   f xss = ys : f (ys : xss) where
     ys = (map sum $ zipWith take [1..] xss) ++ [1]
-- Reinhard Zumkeller, Sep 05 2014

G:=-1+1/product(1-t*x^j,j=1..15): Gser:=simplify(series(G,x=0,17)): for n from 1 to 14 do P[n]:=coeff(Gser,x^n) od: for n from 1 to 14 do seq(coeff(P[n],t^j),j=1..n) od; # yields sequence in triangular form; Emeric Deutsch, Feb 12 2006
with(combstruct):for n from 0 to 18 do seq(count(Partition(n), size=m), m = 1 .. n) od; # Zerinvary Lajos, Mar 30 2009
T := proc(n,k) option remember; if k < 0 or n < 0 then 0 elif k = 0 then if n = 0 then 1 else 0 fi else T(n - 1, k - 1) + T(n - k, k) fi end: seq(print(seq(T(n, k), k=1..n)),n=1..14); # Peter Luschny, Jul 24 2011

Column[Table[ IntegerPartitions[n, {k}] // Length, {n, 1, 20}, {k, 1, n}], Center] (* Frederik Beaujean (beaujean(AT)mpp.mpg.de), Apr 09 2010 *)
(*Recurrence closely related to natural numbers and number of divisors of n*)
Clear[t]; nn = 14; t[n_, 1] = 1; t[n_, k_] := t[n, k] = If[n >= k, Sum[t[n - i, k - 1], {i, 1, n - 1}] - Sum[t[n - i, k], {i, 1, k - 1}], 0];Flatten[Table[Table[t[n, k], {k, 1, n}], {n, 1, nn}]][[1 ;; 96]] (* Mats Granvik, Jan 01 2015 *)
Table[SeriesCoefficient[1/QPochhammer[a q, q], {q, 0, n}, {a, 0, k}], {n, 1, 15}, {k, 1, n}] // Column (* Vladimir Reshetnikov, Nov 18 2016 *)
T[n_, k_] := T[n, k] = If[n>0 && k>0, T[n-1, k-1] + T[n-k, k], Boole[n==0 && k==0]]
Table[T[n, k], {n, 1, 20}, {k, 1, n}] // Flatten (* Robert A. Russell, May 12 2018 after Knuth 7.2.1.4 (39) *)

T(n,k)=#partitions(n-k,k)
for(n=1,9,for(k=1,n,print1(T(n,k)", "))) \\ Charles R Greathouse IV, Jan 04 2016

A8284=[]; A008284(n,k)={for(n=#A8284+1,n,A8284=concat(A8284,[vector(n,k,if(2*k1,A8284[n-k][k]+A8284[n-1][k-1],1),numbpart(n-k)))]));if(k,A8284[n][k],A8284[n])} \\ Without 2nd argument, return row n. - M. F. Hasler, Sep 26 2017

from functools import lru_cache
@lru_cache(maxsize=None)
def A008284_T(n,k):
    if k==n or k==1: return 1
    if k>n: return 0
    return A008284_T(n-1,k-1)+A008284_T(n-k,k) # Chai Wah Wu, Sep 21 2023

from sage.combinat.partition import number_of_partitions_length
[[number_of_partitions_length(n, k) for k in (1..n)] for n in (1..12)] # Peter Luschny, Aug 01 2015

T(n, k) = Sum_{i=1..k} T(n-k, i), for 1 <= k <= n-1; T(n, n) = 1 for n >= 1.
Or, T(n, 1) = T(n, n) = 1, T(n, k) = 0 (k > n), T(n, k) = T(n-1, k-1) + T(n-k, k).
G.f. for k-th column: x^k/(Product_{j=1..k} (1-x^j)). - Wolfdieter Lang, Nov 29 2000
G.f.: A(x, y) = Product_{n>=1} 1/(1-x^n)^(P_n(y)/n), where P_n(y) = Sum_{d|n} eulerphi(n/d)*y^d. - Paul D. Hanna, Jul 13 2004
If k >= n/2, T(n,k) = T(2(n-k),n-k) = A000041(n-k). - Franklin T. Adams-Watters, Jan 12 2006  [Relation included by Hans Loeblich, Apr 16 2019, relation extended by Evan Robinson, Jun 30 2021]
G.f.: G(t,x) = -1 + 1/Product_{j>=1} (1-t*x^j). - Emeric Deutsch, Feb 12 2006
A002865(n) = Sum_{k=2..floor((n+2)/2)} T(n-k+1,k-1). - Reinhard Zumkeller, Nov 04 2007
A000700(n) = Sum_{k=1..n} (-1)^(n-k) T(n,k). - Jeremy L. Martin, Jul 06 2013
G.f.: -1 + e^(F(x,z)), where F(x,z) = Sum_{n >= 1} (x*z)^n/(n*(1 - z^n)) is a g.f. for A126988. - Peter Bala, Jan 13 2015
Also, T(n, n-k) = k for k = 1, 2, 3; n >= 2k. T(n, 2) = floor(n/2). T(n, 3) = round(n^2/12). - M. F. Hasler, Sep 26 2017
T(n,k) = [n>0 & k>0] * (T(n-1,k-1) + T(n-k,k)) + [n==0 & k==0]. - Robert A. Russell, May 12 2018 from Knuth 7.2.1.4 (39)
T(n, k) = Sum_{i=0..n-1} T(n-ik-1, k-1) for k >= 1; T(-n, k) = 0 for n > 0; T(n, 0) = [n==0]. - Joshua Swanson (writing for Juexian Li), May 24 2020

A001399 a(n) is the number of partitions of n into at most 3 parts; also partitions of n+3 in which the greatest part is 3; also number of unlabeled multigraphs with 3 nodes and n edges.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 19, 21, 24, 27, 30, 33, 37, 40, 44, 48, 52, 56, 61, 65, 70, 75, 80, 85, 91, 96, 102, 108, 114, 120, 127, 133, 140, 147, 154, 161, 169, 176, 184, 192, 200, 208, 217, 225, 234, 243, 252, 261, 271, 280, 290, 300, 310, 320, 331, 341

Offset: 0

N. J. A. Sloane

Also number of tripods (trees with exactly 3 leaves) on n vertices. - Eric W. Weisstein, Mar 05 2011
Also number of partitions of n+3 into exactly 3 parts; number of partitions of n in which the greatest part is less than or equal to 3; and the number of nonnegative solutions to b + 2c + 3d = n.
Also a(n) gives number of partitions of n+6 into 3 distinct parts and number of partitions of 2n+9 into 3 distinct and odd parts, e.g., 15 = 11 + 3 + 1 = 9 + 5 + 1 = 7 + 5 + 3. - Jon Perry, Jan 07 2004
Also bracelets with n+3 beads 3 of which are red (so there are 2 possibilities with 5 beads).
More generally, the number of partitions of n into at most k parts is also the number of partitions of n+k into k positive parts, the number of partitions of n+k in which the greatest part is k, the number of partitions of n in which the greatest part is less than or equal to k, the number of partitions of n+k(k+1)/2 into exactly k distinct positive parts, the number of nonnegative solutions to b + 2c + 3d + ... + kz = n and the number of nonnegative solutions to 2c + 3d + ... + kz <= n. - Henry Bottomley, Apr 17 2001
Also coefficient of q^n in the expansion of (m choose 3)_q as m goes to infinity. - Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002
From Winston C. Yang (winston(AT)cs.wisc.edu), Apr 30 2002: (Start)
Write 1,2,3,4,... in a hexagonal spiral around 0, then a(n) for n > 0 is formed by the folding points (including the initial 1). The spiral begins:
.
                 85--84--83--82--81--80
                 /                     \
               86  56--55--54--53--52  79
               /   /                 \   \
             87  57  33--32--31--30  51  78
             /   /   /             \   \   \
           88  58  34  16--15--14  29  50  77
           /   /   /   /         \   \   \   \
         89  59  35  17   5---4  13  28  49  76
         /   /   /   /   /     \   \   \   \   \
       90  60  36  18   6   0   3  12  27  48  75
       /   /   /   /   /   /   /   /   /   /   /
     91  61  37  19   7   1---2  11  26  47  74
           \   \   \   \         /   /   /   /
           62  38  20   8---9--10  25  46  73
             \   \   \             /   /   /
             63  39  21--22--23--24  45  72
               \   \                 /   /
               64  40--41--42--43--44  71
                 \                     /
                 65--66--67--68--69--70
.
a(p) is maximal number of hexagons in a polyhex with perimeter at most 2p + 6. (End)
a(n-3) is the number of partitions of n into 3 distinct parts, where 0 is allowed as a part. E.g., at n=9, we can write 8+1+0, 7+2+0, 6+3+0, 4+5+0, 1+2+6, 1+3+5 and 2+3+4, which is a(6)=7. - Jon Perry, Jul 08 2003
a(n) gives number of partitions of n+6 into parts <=3 where each part is used at least once (subtract 6=1+2+3 from n). - Jon Perry, Jul 03 2004
This is also the number of partitions of n+3 into exactly 3 parts (there is a 1-to-1 correspondence between the number of partitions of n+3 in which the greatest part is 3 and the number of partitions of n+3 into exactly three parts). - Graeme McRae, Feb 07 2005
Apply the Riordan array (1/(1-x^3),x) to floor((n+2)/2). - Paul Barry, Apr 16 2005
Also, number of triangles that can be created with odd perimeter 3,5,7,9,11,... with all sides whole numbers. Note that triangles with even perimeter can be generated from the odd ones by increasing each side by 1. E.g., a(1) = 1 because perimeter 3 can make {1,1,1} 1 triangle. a(4) = 3 because perimeter 9 can make {1,4,4} {2,3,4} {3,3,3} 3 possible triangles. - Bruce Love (bruce_love(AT)ofs.edu.sg), Nov 20 2006
Also number of nonnegative solutions of the Diophantine equation x+2*y+3*z=n, cf. Pólya/Szegő reference.
From Vladimir Shevelev, Apr 23 2011: (Start)
Also a(n-3), n >= 3, is the number of non-equivalent necklaces of 3 beads each of them painted by one of n colors.
The sequence {a(n-3), n >= 3} solves the so-called Reis problem about convex k-gons in case k=3 (see our comment to A032279).
a(n-3) (n >= 3) is an essentially unimprovable upper estimate for the number of distinct values of the permanent in (0,1)-circulants of order n with three 1's in every row. (End)
A001399(n) is the number of 3-tuples (w,x,y) having all terms in {0,...,n} and w = 2*x+3*y. - Clark Kimberling, Jun 04 2012
Also, for n >= 3, a(n-3) is the number of the distinct triangles in an n-gon, see the Ngaokrajang links. - Kival Ngaokrajang, Mar 16 2013
Also, a(n) is the total number of 5-curve coin patterns (5C4S type: 5 curves covering full 4 coins and symmetry) packing into fountain of coins base (n+3). See illustration in links. - Kival Ngaokrajang, Oct 16 2013
Also a(n) = half the number of minimal zero sequences for Z_n of length 3 [Ponomarenko]. - N. J. A. Sloane, Feb 25 2014
Also, a(n) equals the number of linearly-independent terms at 2n-th order in the power series expansion of an Octahedral Rotational Energy Surface (cf. Harter & Patterson). - Bradley Klee, Jul 31 2015
Also Molien series for invariants of finite Coxeter groups D_3 and A_3. - N. J. A. Sloane, Jan 10 2016
Number of different distributions of n+6 identical balls in 3 boxes as x,y,z where 0 < x < y < z. - Ece Uslu and Esin Becenen, Jan 11 2016
a(n) is also the number of partitions of 2*n with <= n parts and no part >= 4. The bijection to partitions of n with no part >= 4 is: 1 <-> 2, 2 <-> 1 + 3, 3 <-> 3 + 3 (observing the order of these rules). The <- direction uses the following fact for partitions of 2*n with <= n parts and no part >=4: for each part 1 there is a part 3, and an even number (including 0) of remaining parts 3. - Wolfdieter Lang, May 21 2019
List of the terms in A000567(n>=1), A049450(n>=1), A033428(n>=1), A049451(n>=1), A045944(n>=1), and A003215(n) in nondecreasing order. List of the numbers A056105(n)-1, A056106(n)-1, A056107(n)-1, A056108(n)-1, A056109(n)-1, and A003215(m) with n >= 1 and m >= 0 in nondecreasing order. Numbers of the forms 3n*(n-1)+1, n*(3n-2), n*(3n-1), 3n^2, n*(3n+1), n*(3n+2) with n >= 1 listed in nondecreasing order. Integers m such that lattice points from 1 through m on a hexagonal spiral starting at 1 forms a convex polygon. - Ya-Ping Lu, Jan 24 2024

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 7*x^6 + 8*x^7 + 10*x^8 + 12*x^9 + ...
Recall that in a necklace the adjacent beads have distinct colors. Suppose we have n colors with labels 1,...,n. Two colorings of the beads are equivalent if the cyclic sequences of the distances modulo n between labels of adjacent colors have the same period. If n=4, all colorings are equivalent. E.g., for the colorings {1,2,3} and {1,2,4} we have the same period {1,1,2} of distances modulo 4. So, a(n-3)=a(1)=1. If n=5, then we have two such periods {1,1,3} and {1,2,2} modulo 5. Thus a(2)=2. - _Vladimir Shevelev_, Apr 23 2011
a(0) = 1, i.e., {1,2,3} Number of different distributions of 6 identical balls to 3 boxes as x,y and z where 0 < x < y < z. - _Ece Uslu_, Esin Becenen, Jan 11 2016
a(3) = 3, i.e., {1,2,6}, {1,3,5}, {2,3,4} Number of different distributions of 9 identical balls in 3 boxes as x,y and z where 0 < x < y < z. - _Ece Uslu_, Esin Becenen, Jan 11 2016
From _Gus Wiseman_, Apr 15 2019: (Start)
The a(0) = 1 through a(8) = 10 integer partitions of n with at most three parts are the following. The Heinz numbers of these partitions are given by A037144.
  ()  (1)  (2)   (3)    (4)    (5)    (6)    (7)    (8)
           (11)  (21)   (22)   (32)   (33)   (43)   (44)
                 (111)  (31)   (41)   (42)   (52)   (53)
                        (211)  (221)  (51)   (61)   (62)
                               (311)  (222)  (322)  (71)
                                      (321)  (331)  (332)
                                      (411)  (421)  (422)
                                             (511)  (431)
                                                    (521)
                                                    (611)
The a(0) = 1 through a(7) = 8 integer partitions of n + 3 whose greatest part is 3 are the following. The Heinz numbers of these partitions are given by A080193.
  (3)  (31)  (32)   (33)    (322)    (332)     (333)      (3322)
             (311)  (321)   (331)    (3221)    (3222)     (3331)
                    (3111)  (3211)   (3311)    (3321)     (32221)
                            (31111)  (32111)   (32211)    (33211)
                                     (311111)  (33111)    (322111)
                                               (321111)   (331111)
                                               (3111111)  (3211111)
                                                          (31111111)
Non-isomorphic representatives of the a(0) = 1 through a(5) = 5 unlabeled multigraphs with 3 vertices and n edges are the following.
  {}  {12}  {12,12}  {12,12,12}  {12,12,12,12}  {12,12,12,12,12}
            {13,23}  {12,13,23}  {12,13,23,23}  {12,13,13,23,23}
                     {13,23,23}  {13,13,23,23}  {12,13,23,23,23}
                                 {13,23,23,23}  {13,13,23,23,23}
                                                {13,23,23,23,23}
The a(0) = 1 through a(8) = 10 strict integer partitions of n - 6 with three parts are the following (A = 10, B = 11). The Heinz numbers of these partitions are given by A007304.
  (321)  (421)  (431)  (432)  (532)  (542)  (543)  (643)   (653)
                (521)  (531)  (541)  (632)  (642)  (652)   (743)
                       (621)  (631)  (641)  (651)  (742)   (752)
                              (721)  (731)  (732)  (751)   (761)
                                     (821)  (741)  (832)   (842)
                                            (831)  (841)   (851)
                                            (921)  (931)   (932)
                                                   (A21)   (941)
                                                           (A31)
                                                           (B21)
The a(0) = 1 through a(8) = 10 integer partitions of n + 3 with three parts are the following. The Heinz numbers of these partitions are given by A014612.
  (111)  (211)  (221)  (222)  (322)  (332)  (333)  (433)  (443)
                (311)  (321)  (331)  (422)  (432)  (442)  (533)
                       (411)  (421)  (431)  (441)  (532)  (542)
                              (511)  (521)  (522)  (541)  (551)
                                     (611)  (531)  (622)  (632)
                                            (621)  (631)  (641)
                                            (711)  (721)  (722)
                                                   (811)  (731)
                                                          (821)
                                                          (911)
The a(0) = 1 through a(8) = 10 integer partitions of n whose greatest part is <= 3 are the following. The Heinz numbers of these partitions are given by A051037.
  ()  (1)  (2)   (3)    (22)    (32)     (33)      (322)      (332)
           (11)  (21)   (31)    (221)    (222)     (331)      (2222)
                 (111)  (211)   (311)    (321)     (2221)     (3221)
                        (1111)  (2111)   (2211)    (3211)     (3311)
                                (11111)  (3111)    (22111)    (22211)
                                         (21111)   (31111)    (32111)
                                         (111111)  (211111)   (221111)
                                                   (1111111)  (311111)
                                                              (2111111)
                                                              (11111111)
The a(0) = 1 through a(6) = 7 strict integer partitions of 2n+9 with 3 parts, all of which are odd, are the following. The Heinz numbers of these partitions are given by A307534.
  (5,3,1)  (7,3,1)  (7,5,1)  (7,5,3)   (9,5,3)   (9,7,3)   (9,7,5)
                    (9,3,1)  (9,5,1)   (9,7,1)   (11,5,3)  (11,7,3)
                             (11,3,1)  (11,5,1)  (11,7,1)  (11,9,1)
                                       (13,3,1)  (13,5,1)  (13,5,3)
                                                 (15,3,1)  (13,7,1)
                                                           (15,5,1)
                                                           (17,3,1)
The a(0) = 1 through a(8) = 10 strict integer partitions of n + 3 with 3 parts where 0 is allowed as a part (A = 10):
  (210)  (310)  (320)  (420)  (430)  (530)  (540)  (640)  (650)
                (410)  (510)  (520)  (620)  (630)  (730)  (740)
                       (321)  (610)  (710)  (720)  (820)  (830)
                              (421)  (431)  (810)  (910)  (920)
                                     (521)  (432)  (532)  (A10)
                                            (531)  (541)  (542)
                                            (621)  (631)  (632)
                                                   (721)  (641)
                                                          (731)
                                                          (821)
The a(0) = 1 through a(7) = 7 integer partitions of n + 6 whose distinct parts are 1, 2, and 3 are the following. The Heinz numbers of these partitions are given by A143207.
  (321)  (3211)  (3221)   (3321)    (32221)    (33221)     (33321)
                 (32111)  (32211)   (33211)    (322211)    (322221)
                          (321111)  (322111)   (332111)    (332211)
                                    (3211111)  (3221111)   (3222111)
                                               (32111111)  (3321111)
                                                           (32211111)
                                                           (321111111)
(End)
Partitions of 2*n with <= n parts and no part >= 4: a(3) = 3 from (2^3), (1,2,3), (3^2) mapping to (1^3), (1,2), (3), the partitions of 3 with no part >= 4, respectively. - _Wolfdieter Lang_, May 21 2019

R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; Chapter III, Problem 33.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 110, D(n); page 263, #18, P_n^{3}.
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 517.
H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 2.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 88, (4.1.18).
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 275.
R. Honsberger, Mathematical Gems III, Math. Assoc. Amer., 1985, p. 39.
J. H. van Lint, Combinatorial Seminar Eindhoven, Lecture Notes Math., 382 (1974), see pp. 33-34.
G. Pólya and G. Szegő, Problems and Theorems in Analysis I (Springer 1924, reprinted 1972), Part One, Chap. 1, Sect. 1, Problem 25.
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).

Marius A. Burtea, Table of n, a(n) for n = 0..17501 (terms 0..1000 from T. D. Noe, terms 14001 onwards corrected by Sean A. Irvine, April 25 2019)
Hamid Afshar, Branislav Cvetkovic, Sabine Ertl, Daniel Grumiller, and Niklas Johansson, Conformal Chern-Simons holography-lock, stock and barrel, arXiv preprint arXiv:1110.5644 [hep-th], 2011.
C. Ahmed, P. Martin, and V. Mazorchuk, On the number of principal ideals in d-tonal partition monoids, arXiv preprint arXiv:1503.06718 [math.CO], 2015.
Nesrine Benyahia-Tani, Zahra Yahi, and Sadek Bouroubi, Ordered and non-ordered non-congruent convex quadrilaterals inscribed in a regular n-gon, Rostocker Math. Kolloq. 68 (2013), 71-79.
N. Benyahia Tani, Z. Yahi, and S. Bouroubi, Ordered and non-ordered non-isometric convex quadrilaterals inscribed in a regular n-gon, Bulletin du Laboratoire Liforce 01 (2014), 1-9.
Jonathan Bloom and Nathan McNew, Counting pattern-avoiding integer partitions, arXiv:1908.03953 [math.CO], 2019.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
S. J. Cyvin, B. N. Cyvin, J. Brunvoll, I. Gutman, Chen Rong-si, S. El-Basil, and Zhang Fuji, Polygonal Systems Including the Corannulene and Coronene Homologs: Novel Applications of Pólya's Theorem, Z. Naturforsch., 52a (1997), 867-873.
Lucia De Luca and Gero Friesecke, Classification of particle numbers with unique Heitmann-Radin minimizer, arXiv:1701.07231 [math-ph], 2017.
Nick Fischer and Christian Ikenmeyer, The Computational Complexity of Plethysm Coefficients, arXiv:2002.00788 [cs.CC], 2020.
H. Gupta, Enumeration of incongruent cyclic k-gons, Indian J. Pure and Appl. Math., 10 (1979), no.8, 964-999.
W. G. Harter and C. W. Patterson, Asymptotic eigensolutions of fourth and sixth rank octahedral tensor operators, Journal of Mathematical Physics, 20.7 (1979), 1453-1459. alternate copy
M. D. Hirschhorn and J. A. Sellers, Enumeration of unigraphical partitions, JIS 11 (2008) 08.4.6.
R. Honsberger, Mathematical Gems III, Math. Assoc. Amer., 1985, p. 39. [Annotated scanned copy]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 352.
J. H. Jordan, R. Walch, and R. J. Wisner, Triangles with integer sides, Amer. Math. Monthly 86 (1979), 686-689.
Alexander V. Karpov, An Informational Basis for Voting Rules, NRU Higher School of Economics. Series WP BRP "Economics/EC". 2018. No. 188.
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Clark Kimberling, A Combinatorial Classification of Triangle Centers on the Line at Infinity, J. Int. Seq., Vol. 22 (2019), Article 19.5.4.
Axel Kleinschmidt and Valentin Verschinin, Tetrahedral modular graph functions, J. High Energy Phys. 2017, No. 9, Paper No. 155, 38 p. (2017), eq (3.40).
Mathematics Stack Exchange, What does "pcr" stand for. [This is Comtet's notation for "prime circulator". See pp. 109-110.]
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Kival Ngaokrajang, Distinct triangles in n-gon for n = 3..9, Distinct triangles in 45-gon
Kival Ngaokrajang, Illustration of 5-curves coins patterns
Andrew N. Norris, Higher derivatives and the inverse derivative of a tensor-valued function of a tensor, arXiv:0707.0115 [math.SP], 2007; Equation 3.28, p. 10.
Jon Perry, More Partition Function.
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.
Vadim Ponomarenko, Minimal zero sequences of finite cyclic groups, INTEGERS 4 (2004), #A24.
Vladimir Shevelev, Necklaces and convex k-gons, Indian J. Pure and Appl. Math. 35(5) (2004), 629-638.
Vladimir Shevelev, Spectrum of permanent's values and its extremal magnitudes in Lambda_n^3 and Lambda_n(alpha,beta,gamma), arXiv:1104.4051 [math.CO], 2011. (Cf. Section 5).
Devansh Singh and S. N. Mishra, Representing K-parts Integer Partitions, Global Journal of Pure and Applied Mathematics (GJPAM), Volume 15 Number 6 (2019), pp. 889-907.
Karl Hermann Struve, Fresnel's Interferenzerscheinungen: Theoretisch und Experimentell Bearbeitet, Dorpat, 1881 (Thesis). [Gives the Round(n^2/12) formula.]
James Tanton, Young students approach integer triangles, FOCUS 22(5) (2002), 4 - 6.
James Tanton, Integer Triangles, Chapter 11 in "Mathematics Galore!" (MAA, 2012).
James Tilley, Stan Wagon, and Eric Weisstein, A Catalog of Facially Complete Graphs, arXiv:2409.11249 [math.CO], 2024. See p. 11.
Richard Vale and Shayne Waldron, The construction of G-invariant finite tight frames, J. Four. Anal. Applic. 22 (2016), 1097-1120.
Eric Weisstein's World of Mathematics, Tripod.
Gus Wiseman, Number of ordered triples of distinct positive integers summing to n.
Gus Wiseman, Number of non-unimodal triples of distinct positive integers summing to n.
Gus Wiseman, Number of unimodal triples of distinct positive integers summing to n.
Winston C. Yang, Maximal and minimal polyhexes, 2002.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).
Index entries for Molien series.
Index entries for two-way infinite sequences

Cf. A008724, A003082, A117485, A026810, A026811, A026812, A026813, A026814, A026815, A026816, A000228, A036496, A008619, A001400, A001401, A069905, A008615, row 3 of A192517.
Molien series for finite Coxeter groups D_3 through D_12 are A001399, A051263, A266744-A266751.
Cf. A007304, A014612, A037144, A051037, A080193, A143207, A307534.

a001399 = p [1,2,3] where
   p _      0 = 1
   p []     _ = 0
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Feb 28 2013

I:=[1,1,2,3,4,5]; [n le 6 select I[n] else Self(n-1)+Self(n-2)-Self(n-4)-Self(n-5)+Self(n-6): n in [1..80]]; // Vincenzo Librandi, Feb 14 2015

[#RestrictedPartitions(n,{1,2,3}): n in [0..62]]; // Marius A. Burtea, Jan 06 2019

[Round((n+3)^2/12): n in [0..70]]; // Marius A. Burtea, Jan 06 2019

A001399 := proc(n)
    round( (n+3)^2/12) ;
end proc:
seq(A001399(n),n=0..40) ;
with(combstruct):ZL4:=[S,{S=Set(Cycle(Z,card<4))}, unlabeled]:seq(count(ZL4,size=n),n=0..61); # Zerinvary Lajos, Sep 24 2007
B:=[S,{S = Set(Sequence(Z,1 <= card),card <=3)},unlabelled]: seq(combstruct[count](B, size=n), n=0..61); # Zerinvary Lajos, Mar 21 2009

CoefficientList[ Series[ 1/((1 - x)*(1 - x^2)*(1 - x^3)), {x, 0, 65} ], x ]
Table[ Length[ IntegerPartitions[n, 3]], {n, 0, 61} ] (* corrected by Jean-François Alcover, Aug 08 2012 *)
k = 3; Table[(Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n + Binomial[If[OddQ[n], n - 1, n - If[OddQ[k], 2, 0]]/2, If[OddQ[k], k - 1, k]/2])/2, {n, k, 50}] (* Robert A. Russell, Sep 27 2004 *)
LinearRecurrence[{1,1,0,-1,-1,1},{1,1,2,3,4,5},70] (* Harvey P. Dale, Jun 21 2012 *)
a[ n_] := With[{m = Abs[n + 3] - 3}, Length[ IntegerPartitions[ m, 3]]]; (* Michael Somos, Dec 25 2014 *)
k=3 (* Number of red beads in bracelet problem *);CoefficientList[Series[(1/k Plus@@(EulerPhi[#] (1-x^#)^(-(k/#))&/@Divisors[k])+(1+x)/(1-x^2)^Floor[(k+2)/2])/2,{x,0,50}],x] (* Herbert Kociemba, Nov 04 2016 *)
Table[Length[Select[IntegerPartitions[n,{3}],UnsameQ@@#&]],{n,0,30}] (* Gus Wiseman, Apr 15 2019 *)

{a(n) = round((n + 3)^2 / 12)}; /* Michael Somos, Sep 04 2006 */

[round((n+3)**2 / 12) for n in range(0,62)] # Ya-Ping Lu, Jan 24 2024

G.f.: 1/((1 - x) * (1 - x^2) * (1 - x^3)) = -1/((x+1)*(x^2+x+1)*(x-1)^3); Simon Plouffe in his 1992 dissertation
a(n) = round((n + 3)^2/12). Note that this cannot be of the form (2*i + 1)/2, so ties never arise.
a(n) = A008284(n+3, 3), n >= 0.
a(n) = 1 + a(n-2) + a(n-3) - a(n-5) for all n in Z. - Michael Somos, Sep 04 2006
a(n) = a(-6 - n) for all n in Z. - Michael Somos, Sep 04 2006
a(6*n) = A003215(n), a(6*n + 1) = A000567(n + 1), a(6*n + 2) = A049450(n + 1), a(6*n + 3) = A033428(n + 1), a(6*n + 4) = A049451(n + 1), a(6*n + 5) = A045944(n + 1).
a(n) = a(n-1) + A008615(n+2) = a(n-2) + A008620(n) = a(n-3) + A008619(n) = A001840(n+1) - a(n-1) = A002620(n+2) - A001840(n) = A000601(n) - A000601(n-1). - Henry Bottomley, Apr 17 2001
P(n, 3) = (1/72) * (6*n^2 - 7 - 9*pcr{1, -1}(2, n) + 8*pcr{2, -1, -1}(3, n)) (see Comtet). [Here "pcr" stands for "prime circulator" and it is defined on p. 109 of Comtet, while the formula appears on p. 110. - Petros Hadjicostas, Oct 03 2019]
Let m > 0 and -3 <= p <= 2 be defined by n = 6*m+p-3; then for n > -3, a(n) = 3*m^2 + p*m, and for n = -3, a(n) = 3*m^2 + p*m + 1. - Floor van Lamoen, Jul 23 2001
72*a(n) = 17 + 6*(n+1)*(n+5) + 9*(-1)^n - 8*A061347(n). - Benoit Cloitre, Feb 09 2003
From Jon Perry, Jun 17 2003: (Start)
a(n) = 6*t(floor(n/6)) + (n%6) * (floor(n/6) + 1) + (n mod 6 == 0?1:0), where t(n) = n*(n+1)/2.
a(n) = ceiling(1/12*n^2 + 1/2*n) + (n mod 6 == 0?1:0).
[Here "n%6" means "n mod 6" while "(n mod 6 == 0?1:0)" means "if n mod 6 == 0 then 1, else 0" (as in C).]
(End)
a(n) = Sum_{i=0..floor(n/3)} 1 + floor((n - 3*i)/2). - Jon Perry, Jun 27 2003
a(n) = Sum_{k=0..n} floor((k + 2)/2) * (cos(2*Pi*(n - k)/3 + Pi/3)/3 + sqrt(3) * sin(2*Pi*(n-k)/3 + Pi/3)/3 + 1/3). - Paul Barry, Apr 16 2005
(m choose 3)_q = (q^m-1) * (q^(m-1) - 1) * (q^(m-2) - 1)/((q^3 - 1) * (q^2 - 1) * (q - 1)).
a(n) = Sum_{k=0..floor(n/2)} floor((3 + n - 2*k)/3). - Paul Barry, Nov 11 2003
A117220(n) = a(A003586(n)). - Reinhard Zumkeller, Mar 04 2006
a(n) = 3 * Sum_{i=2..n+1} floor(i/2) - floor(i/3). - Thomas Wieder, Feb 11 2007
Identical to the number of points inside or on the boundary of the integer grid of {I, J}, bounded by the three straight lines I = 0, I - J = 0 and I + 2J = n. - Jonathan Vos Post, Jul 03 2007
a(n) = A026820(n,3) for n > 2. - Reinhard Zumkeller, Jan 21 2010
Euler transform of length 3 sequence [ 1, 1, 1]. - Michael Somos, Feb 25 2012
a(n) = A005044(2*n + 3) = A005044(2*n + 6). - Michael Somos, Feb 25 2012
a(n) = A000212(n+3) - A002620(n+3). - Richard R. Forberg, Dec 08 2013
a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6). - David Neil McGrath, Feb 14 2015
a(n) = floor((n^2+3)/12) + floor((n+2)/2). - Giacomo Guglieri, Apr 02 2019
From Devansh Singh, May 28 2020: (Start)
Let p(n, 3) be the number of 3-part integer partitions in which every part is > 0.
Then for n >= 3, p(n, 3) is equal to:
  (n^2 - 1)/12 when n is odd and 3 does not divide n.
  (n^2 + 3)/12 when n is odd and 3 divides n.
  (n^2 - 4)/12 when n is even and 3 does not divide n.
  (n^2)/12 when n is even and 3 divides n.
For n >= 3, p(n, 3) = a(n-3). (End)
a(n) = floor(((n+3)^2 + 4)/12). - Vladimír Modrák, Zuzana Soltysova, Dec 08 2020
Sum_{n>=0} 1/a(n) = 15/4 - Pi/(2*sqrt(3)) + Pi^2/18 + tanh(Pi/(2*sqrt(3)))*Pi/sqrt(3). - Amiram Eldar, Sep 29 2022
E.g.f.: exp(-x)*(9 + exp(2*x)*(47 + 42*x + 6*x^2) + 16*exp(x/2)*cos(sqrt(3)*x/2))/72. - Stefano Spezia, Mar 05 2023
a(6n) = 1+6*A000217(n); Sum_{i=1..n} a(6*i) = A000578(n+1). - David García Herrero, May 05 2024

Name edited by Gus Wiseman, Apr 15 2019

A026810 Number of partitions of n in which the greatest part is 4.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 3, 5, 6, 9, 11, 15, 18, 23, 27, 34, 39, 47, 54, 64, 72, 84, 94, 108, 120, 136, 150, 169, 185, 206, 225, 249, 270, 297, 321, 351, 378, 411, 441, 478, 511, 551, 588, 632, 672, 720, 764, 816, 864, 920, 972, 1033, 1089, 1154, 1215, 1285, 1350

Offset: 0

Clark Kimberling

Also number of partitions of n into exactly 4 parts.
Also the number of weighted cubic graphs on 4 nodes (=the tetrahedron) with weight n. - R. J. Mathar, Nov 03 2018
From Gus Wiseman, Jun 27 2021: (Start)
Also the number of strict integer partitions of 2n with alternating sum 4, or (by conjugation) partitions of 2n covering an initial interval of positive integers with exactly 4 odd parts. The strict partitions with alternating sum 4 are:
  (4)  (5,1)  (6,2)    (7,3)    (8,4)      (9,5)      (10,6)
              (5,2,1)  (5,3,2)  (5,4,3)    (6,5,3)    (7,6,3)
                       (6,3,1)  (6,4,2)    (7,5,2)    (8,6,2)
                                (7,4,1)    (8,5,1)    (9,6,1)
                                (6,3,2,1)  (6,4,3,1)  (6,5,4,1)
                                           (7,4,2,1)  (7,4,3,2)
                                                      (7,5,3,1)
                                                      (8,5,2,1)
                                                      (6,4,3,2,1)
(End)

			From _Gus Wiseman_, Jun 27 2021: (Start)
The a(4) = 1 through a(10) = 9 partitions of length 4:
  (1111)  (2111)  (2211)  (2221)  (2222)  (3222)  (3322)
                  (3111)  (3211)  (3221)  (3321)  (3331)
                          (4111)  (3311)  (4221)  (4222)
                                  (4211)  (4311)  (4321)
                                  (5111)  (5211)  (4411)
                                          (6111)  (5221)
                                                  (5311)
                                                  (6211)
                                                  (7111)
(End)

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 275.
D. E. Knuth, The Art of Computer Programming, vol. 4,fascicle 3, Generating All Combinations and Partitions, Section 7.2.1.4., p. 56, exercise 31.

Washington Bomfim, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-2,0,0,1,1,-1).

Cf. A001400, A026811, A026812, A026813, A026814, A026815, A026816, A069905 (3 positive parts), A002621 (partial sums), A005044 (first differences).
A non-strict version is A000710 or A088218.
This is column k = 2 of A152146.
A reverse version is A343941.
Cf. A000041, A000070, A000097, A067659, A103919, A120452, A236559, A239830, A306145, A343942, A344616, A344649, A344651.

[Round((n^3+3*n^2-9*n*(n mod 2))/144): n in [0..60]]; // Vincenzo Librandi, Oct 14 2015

A049347 := proc(n)
        op(1+(n mod 3),[1,-1,0]) ;
end proc:
A056594 := proc(n)
        op(1+(n mod 4),[1,0,-1,0]) ;
end proc:
A026810 := proc(n)
        1/288*(n+1)*(2*n^2+4*n-13+9*(-1)^n) ;
        %-A049347(n)/9 ;
        %+A056594(n)/8 ;
end proc: # R. J. Mathar, Jul 03 2012

Table[Count[IntegerPartitions[n], {4, _}], {n, 0, 60}]
LinearRecurrence[{1, 1, 0, 0, -2, 0, 0, 1, 1, -1}, {0, 0, 0, 0, 1, 1, 2, 3, 5, 6}, 60] (* Vincenzo Librandi, Oct 14 2015 *)
Table[Length[IntegerPartitions[n, {4}]], {n, 0, 60}] (* Eric Rowland, Mar 02 2017 *)
CoefficientList[Series[x^4/Product[1 - x^k, {k, 1, 4}], {x, 0, 60}], x] (* Robert A. Russell, May 13 2018 *)

for(n=0, 60, print(n, " ", round((n^3 + 3*n^2 -9*n*(n % 2))/144))); \\ Washington Bomfim, Jul 03 2012

x='x+O('x^60); concat([0, 0, 0, 0], Vec(x^4/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)))) \\ Altug Alkan, Oct 14 2015

vector(60, n, n--; (n+1)*(2*n^2+4*n-13+9*(-1)^n)/288 + real(I^n)/8 - ((n+2)%3-1)/9) \\ Altug Alkan, Oct 26 2015

print1(0,", "); for(n=1,60,j=0;forpart(v=n,j++,,[4,4]); print1(j,", ")) \\ Hugo Pfoertner, Oct 01 2018

G.f.: x^4/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) = x^4/((1-x)^4*(1+x)^2*(1+x+x^2)*(1+x^2)).
a(n+4) = A001400(n). - Michael Somos, Apr 07 2012
a(n) = round( (n^3 + 3*n^2 -9*n*(n mod 2))/144 ). - Washington Bomfim, Jan 06 2021 and Jul 03 2012
a(n) = (n+1)*(2*n^2+4*n-13+9*(-1)^n)/288 -A049347(n)/9 +A056594(n)/8. - R. J. Mathar, Jul 03 2012
From Gregory L. Simay, Oct 13 2015: (Start)
a(n) = (n^3 + 3*n^2 - 9*n)/144 + a(m) - (m^3 + 3*m^2 - 9*m)/144 if n = 12k + m and m is odd. For example, a(23) = a(12*1 + 11) = (23^3 + 3*23^2 - 9*23)/144 + a(11) - (11^3 + 3*11^2 - 9*11)/144 = 94.
a(n) = (n^3 + 3*n^2)/144 + a(m) - (m^3 + 3*m^2)/144 if n = 12k + m and m is even. For example, a(22) = a(12*1 + 10) = (22^3 + 3*22^2)/144 + a(10) - (10^3 + 3*10^2)/144 = 84. (End)
a(n) = A008284(n,4). - Robert A. Russell, May 13 2018
From Gregory L. Simay, Jul 28 2019: (Start)
a(2n+1) = a(2n) + a(n+1) - a(n-3) and
a(2n) = a(2n-1) + a(n+2) - a(n-2). (End)

A026811 Number of partitions of n in which the greatest part is 5.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 10, 13, 18, 23, 30, 37, 47, 57, 70, 84, 101, 119, 141, 164, 192, 221, 255, 291, 333, 377, 427, 480, 540, 603, 674, 748, 831, 918, 1014, 1115, 1226, 1342, 1469, 1602, 1747, 1898, 2062, 2233, 2418, 2611, 2818, 3034, 3266, 3507, 3765

Offset: 0

Clark Kimberling

Essentially same as A001401: five zeros followed by A001401.

Also number of partitions of n into exactly 5 parts.

D. E. Knuth, The Art of Computer Programming, vol. 4, fascicle 3, Generating All Combinations and Partitions, Section 7.2.1.4., p. 56, exercise 31.

Washington Bomfim, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-1,-1,-1,1,1,1,0,0,-1,-1,1).

Cf. A026810, A026812, A026813, A026814, A026815, A026816, A002622 (partial sums), A008667 (first differences).

List([0..70],n->NrPartitions(n,5)); # Muniru A Asiru, May 17 2018

Table[Count[IntegerPartitions[n], {5, _}], {n, 0, 55}] (* corrected by Harvey P. Dale, Oct 24 2011 *)
Table[Length[IntegerPartitions[n, {5}]], {n, 0, 55}] (* Eric Rowland, Mar 02 2017 *)
CoefficientList[Series[x^5/Product[1 - x^k, {k, 1, 5}], {x, 0, 65}], x] (* Robert A. Russell, May 13 2018 *)
Drop[LinearRecurrence[{1,1,0,0,-1,-1,-1,1,1,1,0,0,-1,-1,1}, Append[Table[0,{14}],1],110],9] (* Robert A. Russell, May 17 2018 *)

a(n)=round((n^4+10*(n^3+n^2)-75*n-45*(-1)^n*n)/2880);
for(n=0,10000,print(n," ",a(n))); /* b-file format */
/* Washington Bomfim, Jul 03 2012 */

x='x+O('x^99); concat(vector(5), Vec(x^5/prod(k=1, 5, 1-x^k))) \\ Altug Alkan, May 17 2018

a(n) = round( ((n^4+10*(n^3+n^2)-75*n -45*n*(-1)^n)) / 2880 ). - Washington Bomfim, Jul 03 2012
G.f.: x^5/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)). - Joerg Arndt, Jul 04 2012
a(n) = A008284(n,5). - Robert A. Russell, May 13 2018
From Gregory L. Simay, Jul 28 2019: (Start)
a(2n)   = a(2n-1) + a(n+1) + a(n) - a(n-3) - a(n-4);
a(2n+1) = a(2n)   + a(n+3) - a(n-5). (End)
From R. J. Mathar, Jun 23 2021: (Start)
a(n) - a(n-5) = A001400(n-5).
a(n) - a(n-4) = A008669(n-5).
a(n) - a(n-3) = A029007(n-5).
a(n) - a(n-2) = A029032(n-5).
a(n) = +a(n-1) +a(n-2) -a(n-5) -a(n-6) -a(n-7) +a(n-8) +a(n-9) +a(n-10) -a(n-13) -a(n-14) +a(n-15). (End)

More terms from Robert G. Wilson v, Jan 11 2002

a(0)=0 inserted by Joerg Arndt, Jul 04 2012

A026816 Number of partitions of n in which the greatest part is 10.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 55, 75, 97, 128, 164, 212, 267, 340, 423, 530, 653, 807, 984, 1204, 1455, 1761, 2112, 2534, 3015, 3590, 4242, 5013, 5888, 6912, 8070, 9418, 10936, 12690, 14663, 16928, 19466

Offset: 0

Clark Kimberling

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 1..1000 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (1, 1, 0, 0, -1, 0, -1, 0, 0, 0, -1, 1, 1, 1, 2, 0, 0, -1, -1, -1, -1, -3, 0, 0, 1, 1, 2, 2, 1, 1, 0, 0, -3, -1, -1, -1, -1, 0, 0, 2, 1, 1, 1, -1, 0, 0, 0, -1, 0, -1, 0, 0, 1, 1, -1).

Essentially same as A008639.

Cf. A026810, A026811, A026812, A026813, A026814, A026815.

List([0..70],n->NrPartitions(n,10)); # Muniru A Asiru, May 17 2018

[#Partitions(k, 10): k in [1..51]]; // Marius A. Burtea, Jul 13 2019

Table[ Length[ Select[ Partitions[n], First[ # ] == 10 & ]], {n, 1, 60} ]
CoefficientList[Series[x^10/((1 - x) (1 - x^2) (1 - x^3) (1 - x^4) (1 - x^5) (1 - x^6) (1 - x^7) (1 - x^8) (1 - x^9) (1 - x^10)), {x, 0, 60}], x] (* Vincenzo Librandi, Oct 18 2013 *)

concat(vector(9),Vec(1/prod(k=1,10,1-x^k)+O(x^90))) \\ Charles R Greathouse IV, May 06 2015

G.f.: x^10 / (Product_{k=1..10} 1-x^k ). - Colin Barker, Feb 22 2013
a(n) = A008284(n,10). - Robert A. Russell, May 13 2018
a(n) = Sum_{r=1..floor(n/10)} Sum_{q=r..floor((n-r)/9)} Sum_{p=q..floor((n-q-r)/8)} Sum_{o=p..floor((n-p-q-r)/7)} Sum_{m=o..floor((n-o-p-q-r)/6)} Sum_{l=m..floor((n-m-o-p-q-r)/5)} Sum_{k=l..floor((n-l-m-o-p-q-r)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q-r)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q-r)/2)} 1. - Wesley Ivan Hurt, Jul 13 2019

a(0)=0 prepended by Seiichi Manyama, Jun 08 2017

A026813 Number of partitions of n in which the greatest part is 7.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 21, 28, 38, 49, 65, 82, 105, 131, 164, 201, 248, 300, 364, 436, 522, 618, 733, 860, 1009, 1175, 1367, 1579, 1824, 2093, 2400, 2738, 3120, 3539, 4011, 4526, 5102, 5731, 6430, 7190, 8033, 8946, 9953, 11044, 12241

Offset: 0

Clark Kimberling

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 1..1000 from Vincenzo Librandi)
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-1,0,-1,-1,0,1,1,2,0,0,0,-2,-1,-1,0,1,1,0,1,0,0,-1,-1,1).

Cf. A008284, A008636.

Cf. A026810, A026811, A026812, A026814, A026815, A026816.

List([0..70],n->NrPartitions(n,7)); # Muniru A Asiru, May 17 2018

[#Partitions(n,7): n in [0..53]]; // Marius A. Burtea, Jul 01 2019

Table[ Length[ Select[ Partitions[n], First[ # ] == 7 & ]], {n, 1, 60} ]
CoefficientList[Series[x^7/((1 - x) (1 - x^2) (1 - x^3) (1 - x^4) (1 - x^5) (1 - x^6) (1 - x^7)), {x, 0, 60}], x] (* Vincenzo Librandi, Oct 18 2013 *)
Drop[LinearRecurrence[{1,1,0,0,-1,0,-1,-1,0,1,1,2,0,0,0,-2,-1,-1,0,1,1,0,1,0,0,-1,-1,1}, Append[Table[0,{27}],1],121],20] (* Robert A. Russell, May 17 2018 *)

my(x='x+O('x^99)); concat(vector(7), Vec(x^7/prod(k=1, 7, 1-x^k))) \\ Altug Alkan, May 17 2018

G.f.: x^7 / ((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)*(1-x^7)). - Colin Barker, Feb 22 2013
a(n) = A008284(n,7). - Robert A. Russell, May 13 2018
a(n) = A008636(n-7). - R. J. Mathar, Feb 13 2019
a(n) = Sum_{o=1..floor(n/7)} Sum_{m=o..floor((n-o)/6)} Sum_{l=m..floor((n-m-o)/5)} Sum_{k=l..floor((n-l-m-o)/4)} Sum_{j=k..floor((n-k-l-m-o)/3)} Sum_{i=j..floor((n-j-k-l-m-o)/2)} 1. - Wesley Ivan Hurt, Jun 30 2019

More terms from Robert G. Wilson v, Jan 11 2002

a(0)=0 prepended by Seiichi Manyama, Jun 08 2017

A026814 Number of partitions of n in which the greatest part is 8.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 40, 52, 70, 89, 116, 146, 186, 230, 288, 352, 434, 525, 638, 764, 919, 1090, 1297, 1527, 1801, 2104, 2462, 2857, 3319, 3828, 4417, 5066, 5812, 6630, 7564, 8588, 9749, 11018, 12450, 14012, 15765, 17674

Offset: 0

Clark Kimberling

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 1..1000 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (1, 1, 0, 0, -1, 0, -1, 0, -1, 0, 1, 2, 1, 0, 1, -1, -1, -2, -1, -1, 1, 0, 1, 2, 1, 0, -1, 0, -1, 0, -1, 0, 0, 1, 1, -1).

Cf. A026810, A026811, A026812, A026813, A026815, A026816.

List([0..70],n->NrPartitions(n,8)); # Muniru A Asiru, May 17 2018

CoefficientList[Series[x^8/((1 - x) (1 - x^2) (1 - x^3) (1 - x^4) (1 - x^5) (1 - x^6) (1 - x^7) (1 - x^8)), {x, 0, 60}], x] (* Vincenzo Librandi, Oct 18 2013 *)
Table[Count[IntegerPartitions[n],?(Max[#]==8&)],{n,0,55}] (* _Harvey P. Dale, Dec 04 2022 *)

x='x+O('x^99); concat(vector(8), Vec(x^8/prod(k=1, 8, 1-x^k))) \\ Altug Alkan, May 17 2018

G.f.: x^8 / ((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)*(1-x^7)*(1-x^8)). [Colin Barker, Feb 22 2013]
a(n) = A008284(n,8). - Robert A. Russell, May 13 2018
a(n) = Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/2)} 1. - Wesley Ivan Hurt, Jul 04 2019

More terms from Robert G. Wilson v, Jan 11 2002
a(0)=0 prepended by Seiichi Manyama, Jun 08 2017
Two inoperative Mathematica programs deleted by Harvey P. Dale, Dec 04 2022

A026812 Number of partitions of n in which the greatest part is 6.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 14, 20, 26, 35, 44, 58, 71, 90, 110, 136, 163, 199, 235, 282, 331, 391, 454, 532, 612, 709, 811, 931, 1057, 1206, 1360, 1540, 1729, 1945, 2172, 2432, 2702, 3009, 3331, 3692, 4070, 4494, 4935, 5427, 5942, 6510, 7104, 7760

Offset: 0

Clark Kimberling

Also number of partitions of n into 6 parts. - Washington Bomfim, Jan 15 2021

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 1..1000 from Vincenzo Librandi)
G. E. Andrews, Partitions: At the Interface of q-Series and Modular Forms, The Ramanujan Journal 7, 385-400 (2003), Eq.(3.10).
Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-1,0,-2,0,1,1,1,1,0,-2,0,-1,0,0,1,1,-1).

Essentially same as A001402.
Cf. A026810, A026811, A026813, A026814, A026815, A026816.
Cf. A001401, A001402.

List([0..70],n->NrPartitions(n,6)); # Muniru A Asiru, May 17 2018

Table[ Length[ Select[ Partitions[n], First[ # ] == 6 & ]], {n, 1, 60} ]
CoefficientList[Series[x^6/((1 - x) (1 - x^2) (1 - x^3) (1 - x^4) (1 - x^5) (1 - x^6)), {x, 0, 60}], x] (* Vincenzo Librandi, Oct 18 2013 *)
Drop[LinearRecurrence[{1,1,0,0,-1,0,-2,0,1,1,1,1,0,-2,0,-1,0,0,1,1,-1}, Append[Table[0,{20}],1],115],14] (* Robert A. Russell, May 17 2018 *)

my(x='x+O('x^99)); concat(vector(6), Vec(x^6/prod(k=1, 6, 1-x^k))) \\ Altug Alkan, May 17 2018

a = vector(60,n,n--; round((n+11)*((6*n^4+249*n^3+2071*n^2 -4931*n+40621) /518400 +n\2*(n+10)/192+((n+1)\3+n\3*2)/54))); a = concat([0,0,0,0,0,0], a) \\ Washington Bomfim, Jan 16 2021

G.f.: x^6 / ((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)). - Colin Barker, Dec 20 2012
a(n) = A008284(n,6). - Robert A. Russell, May 13 2018
a(n) = Sum_{m=1..floor(n/6)} Sum_{l=m..floor((n-m)/5)} Sum_{k=l..floor((n-l-m)/4)} Sum_{j=k..floor((n-k-l-m)/3)} Sum_{i=j..floor((n-j-k-l-m)/2)} 1. - Wesley Ivan Hurt, Jun 29 2019
a(n) = A001402(n) - A001401(n). a(n) = A001402(n-6). - Washington Bomfim, Jan 15 2021
a(n) = round((1/86400)*n^5 + (1/3840)*n^4 + (19/12960)*n^3 - (n mod 2)*(1/384)*n^2 + (1/17280)*b(n mod 6)*n), where b(0)=96, b(1)=b(5)=-629, b(2)=b(4)=-224, and b(3)=-309. - Washington Bomfim and Jon E. Schoenfield, Jan 16 2021

More terms from Robert G. Wilson v, Jan 11 2002

a(0)=0 prepended by Seiichi Manyama, Jun 08 2017

A288255 Number of nonagons that can be formed with perimeter n.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 14, 21, 28, 39, 50, 69, 87, 116, 145, 189, 233, 299, 363, 458, 553, 687, 820, 1009, 1195, 1453, 1709, 2058, 2404, 2872, 3331, 3948, 4557, 5361, 6152, 7194, 8215, 9547, 10853, 12543, 14199, 16329, 18407, 21067, 23666, 26964, 30179, 34248, 38207

Offset: 9

Seiichi Manyama, Jun 07 2017

Number of (a1, a2, ... , a9) where 1 <= a1 <= ... <= a9 and a1 + a2 + ... + a8 > a9.

Seiichi Manyama, Table of n, a(n) for n = 9..10000
G. E. Andrews, P. Paule and A. Riese, MacMahon's Partition Analysis IX: k-gon partitions, Bull. Austral Math. Soc., 64 (2001), 321-329.
Index entries for linear recurrences with constant coefficients, signature (0, 1, 0, 1, 0, 0, 0, 0, 1, -1, -1, 0, -1, -1, 0, 0, 0, -1, 1, 0, 0, 1, 1, 2, 0, 1, 1, 0, 0, 1, -1, -1, -2, -1, -1, -2, 0, -1, -1, -1, 1, 1, 1, 0, 2, 1, 1, 2, 1, 1, -1, 0, 0, -1, -1, 0, -2, -1, -1, 0, 0, -1, 1, 0, 0, 0, 1, 1, 0, 1, 1, -1, 0, 0, 0, 0, -1, 0, -1, 0, 1).

Number of k-gons that can be formed with perimeter n: A005044 (k=3), A062890 (k=4), A069906 (k=5), A069907 (k=6), A288253 (k=7), A288254 (k=8), this sequence (k=9), A288256 (k=10).

G.f.: x^9/((1-x)*(1-x^2)* ... *(1-x^9)) - x^16/(1-x) * 1/((1-x^2)*(1-x^4)* ... *(1-x^16)).

a(2*n+16) = A026815(2*n+16) - A288343(n), a(2*n+17) = A026815(2*n+17) - A288343(n) for n >= 0. - Seiichi Manyama, Jun 08 2017

A060028 Expansion of (1-x-x^N)/((1-x)(1-x^2)(1-x^3)...(1-x^N)) for N = 9.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 4, 7, 7, 10, 11, 16, 16, 22, 23, 29, 29, 36, 34, 41, 37, 40, 32, 32, 14, 6, -22, -44, -90, -130, -203, -270, -378, -487, -642, -803, -1027, -1260, -1568, -1899, -2320, -2774, -3342, -3955, -4706, -5526, -6507, -7579, -8854, -10243, -11872, -13656

Offset: 0

N. J. A. Sloane, Mar 17 2001

Difference of the number of partitions of n+8 into 8 parts and the number of partitions of n+8 into 9 parts. - Wesley Ivan Hurt, Apr 16 2019

Ray Chandler, Table of n, a(n) for n = 0..1000
P. A. MacMahon, Perpetual reciprocants, Proc. London Math. Soc., 17 (1886), 139-151; Coll. Papers II, pp. 584-596.
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1, 1, 0, 0, -1, 0, -1, 0, 0, -1, 0, 2, 1, 1, 1, 0, -1, -1, -1, -2, -1, -1, 1, 1, 2, 1, 1, 1, 0, -1, -1, -1, -2, 0, 1, 0, 0, 1, 0, 1, 0, 0, -1, -1, 1).

Cf. A026814, A026815.

Cf. For other values of N: A060022 (N=3), A060023 (N=4), A060024 (N=5), A060025 (N=6), A060026 (N=7), A060027 (N=8), this sequence (N=9), A060029 (N=10).

With[{den=Times@@Table[(1-x^n),{n,9}]},CoefficientList[Series[(1-x-x^9)/ den,{x,0,60}],x]] (* Harvey P. Dale, May 22 2012 *)

a(n) = A026814(n+8) - A026815(n+8). - Wesley Ivan Hurt, Apr 16 2019

cp's OEIS Frontend

A008284 Triangle of partition numbers: T(n,k) = number of partitions of n in which the greatest part is k, 1 <= k <= n. Also number of partitions of n into k positive parts, 1 <= k <= n.

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A001399 a(n) is the number of partitions of n into at most 3 parts; also partitions of n+3 in which the greatest part is 3; also number of unlabeled multigraphs with 3 nodes and n edges.

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A026810 Number of partitions of n in which the greatest part is 4.

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A026811 Number of partitions of n in which the greatest part is 5.

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A026816 Number of partitions of n in which the greatest part is 10.

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