A008636 -id:A008636 - cp's OEIS frontend

A026820 Euler's table: triangular array T read by rows, where T(n,k) = number of partitions in which every part is <= k for 1 <= k <= n. Also number of partitions of n into at most k parts.

1, 1, 2, 1, 2, 3, 1, 3, 4, 5, 1, 3, 5, 6, 7, 1, 4, 7, 9, 10, 11, 1, 4, 8, 11, 13, 14, 15, 1, 5, 10, 15, 18, 20, 21, 22, 1, 5, 12, 18, 23, 26, 28, 29, 30, 1, 6, 14, 23, 30, 35, 38, 40, 41, 42, 1, 6, 16, 27, 37, 44, 49, 52, 54, 55, 56, 1, 7, 19, 34, 47, 58, 65, 70, 73, 75, 76, 77

Offset: 1

Clark Kimberling

			Triangle starts:
  1;
  1, 2;
  1, 2,  3;
  1, 3,  4,  5;
  1, 3,  5,  6,  7;
  1, 4,  7,  9, 10, 11;
  1, 4,  8, 11, 13, 14, 15;
  1, 5, 10, 15, 18, 20, 21, 22;
  1, 5, 12, 18, 23, 26, 28, 29, 30;
  1, 6, 14, 23, 30, 35, 38, 40, 41, 42;
  1, 6, 16, 27, 37, 44, 49, 52, 54, 55, 56;
  ...

G. Chrystal, Algebra, Vol. II, p. 558.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIV.2, p. 493.

Alois P. Heinz, Robert G. Wilson v, Rows n = 1..141, 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]
Carolyn Echter, Georg Maier, Juan-Diego Urbina, Caio Lewenkopf, and Klaus Richter, Many-body density of states of bosonic and fermionic gases: a combinatorial approach, arXiv:2409.08696 [cond-mat.quant-gas], 2024. See p. 10.
L. Euler, Introductio in Analysin Infinitorum, Book I, chapter XVI.
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.
R. Sulzgruber, The Symmetry of the q,t-Catalan Numbers, Masterarbeit, Univ. Wien, 2013.
Sergei Viznyuk, C program.
Sergei Viznyuk, Local copy of C program.
Eric Weisstein's World of Mathematics, Partition Function q.
Index entries for sequences related to partitions - _Reinhard Zumkeller_, Jan 21 2010

Partial sums of rows of A008284, row sums give A058397, central terms give A171985, mirror is A058400.
T(n,n) = A000041(n), T(n,1) = A000012(n), T(n,2) = A008619(n) for n>1, T(n,3) = A001399(n) for n>2, T(n,4) = A001400(n) for n>3, T(n,5) = A001401(n) for n>4, T(n,6) = A001402(n) for n>5, T(n,7) = A008636(n) for n>6, T(n,8) = A008637(n) for n>7, T(n,9) = A008638(n) for n>8, T(n,10) = A008639(n) for n>9, T(n,11) = A008640(n) for n>10, T(n,12) = A008641(n) for n>11, T(n,n-2) = A007042(n-1) for n>2, T(n,n-1) = A000065(n) for n>1.
Cf. A008284, A026840, A134737, A173519.

import Data.List (inits)
a026820 n k = a026820_tabl !! (n-1) !! (k-1)
a026820_row n = a026820_tabl !! (n-1)
a026820_tabl = zipWith
   (\x -> map (p x) . tail . inits) [1..] $ tail $ inits [1..] where
   p 0 _ = 1
   p _ [] = 0
   p m ks'@(k:ks) = if m < k then 0 else p (m - k) ks' + p m ks
-- Reinhard Zumkeller, Dec 18 2013

T:= proc(n, k) option remember;
      `if`(n=0 or k=1, 1, T(n, k-1) + `if`(k>n, 0, T(n-k, k)))
    end:
seq(seq(T(n, k), k=1..n), n=1..12); # Alois P. Heinz, Apr 21 2012

t[n_, k_] := Length@ IntegerPartitions[n, k]; Table[ t[n, k], {n, 12}, {k, n}] // Flatten
(* Second program: *)
T[n_, k_] := T[n, k] = If[n==0 || k==1, 1, T[n, k-1] + If[k>n, 0, T[n-k, k]]]; Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 22 2015, after Alois P. Heinz *)

T(n,k)=my(s); forpart(v=n,s++,,k); s \\ Charles R Greathouse IV, Feb 27 2018

from sage.combinat.partition import number_of_partitions_length
from itertools import accumulate
for n in (1..11):
    print(list(accumulate([number_of_partitions_length(n, k) for k in (1..n)])))
# Peter Luschny, Jul 28 2022

T(T(n,n),n) = A134737(n). - Reinhard Zumkeller, Nov 07 2007
T(A000217(n),n) = A173519(n). - Reinhard Zumkeller, Feb 20 2010
T(n,k) = T(n,k-1) + T(n-k,k). - Thomas Dybdahl Ahle, Jun 13 2011
T(n,k) = Sum_{i=1..min(k,floor(n/2))} T(n-i,i) + Sum_{j=1+floor(n/2)..k} A000041(n-j). - Bob Selcoe, Aug 22 2014 [corrected by Álvar Ibeas, Mar 15 2018]
O.g.f.: Product_{i>=0} 1/(1-y*x^i). - Geoffrey Critzer, Mar 11 2012
T(n,k) = A008284(n+k,k). - Álvar Ibeas, Jan 06 2015

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

A008637 Number of partitions of n into at most 8 parts.

Original entry on oeis.org

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, 19805, 22122

Offset: 0

N. J. A. Sloane

For n>7: also number of partitions of n into parts <= 8: a(n)=A026820(n,8). - Reinhard Zumkeller, Jan 21 2010
Molien series for finite Coxeter group of type A_8.
Number of different distributions of n+36 identical balls in 8 boxes as x,y,z,p,q,m,n,h where 0 < x < y < z < p < q < m < n < h. - Ece Uslu and Esin Becenen, Jan 11 2016

			There are a(9)=29 partitions of 9 into parts less than or equal to 8. These are (81)(72)(711)(63)(621)(6111)(54)(531)(522)(5211)(51111)(441)(432)(4311)(4221)(42111)(411111)(333)(3321)(33111)(3222)(32211)(321111)(3111111)(22221)(222111)(2211111)(21111111)(111111111). - _David Neil McGrath_, Apr 14 2015
a(3) = 3, i.e., {1,2,3,4,5,7,8,9}, {1,2,3,4,5,6,8,10}, {1,2,3,4,5,6,7,11}: number of different distributions of 39 identical balls in 8 boxes as x,y,z,p,q,m,n,h where 0 < x < y < z < p < q < m < n < h. - _Ece Uslu_, Esin Becenen, Jan 11 2016

A. Cayley, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 10, p. 415.
H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 2.

T. D. Noe, Table of n, a(n) for n = 0..1000
Tani Akinari, Formula for a(n)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 357
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. A008284.

Strictly different from A008631, although they have similar descriptions.

1/(1-x)/(1-x^2)/(1-x^3)/(1-x^4)/(1-x^5)/(1-x^6)/(1-x^7)/(1-x^8)
with(combstruct):ZL9:=[S,{S=Set(Cycle(Z,card<9))}, unlabeled]:seq(count(ZL9,size=n),n=0..47); # Zerinvary Lajos, Sep 24 2007
B:=[S,{S = Set(Sequence(Z,1 <= card),card <=8)},unlabelled]: seq(combstruct[count](B, size=n), n=0..47); # Zerinvary Lajos, Mar 21 2009

CoefficientList[ Series[ 1/ Product[ 1 - x^n, {n, 1, 8} ], {x, 0, 60} ], x ]

a(n):=floor((-1)^n*((n+1)*(-1)^floor((n+2)/3)+(2*n+3)*(-1)^floor((n+1)/3)+(n+2)*(-1)^floor(n/3))/972+(n+2)*((-1)^n+1)*(-1)^(n/2)/512+(n+18)*(6*n^6+648*n^5+27018*n^4+545616*n^3+5481213*n^2+25163028*n+39226571)/1219276800+(n+1)*(n^2+53*n+826)*(-1)^n/36864+1/2); /* Tani Akinari, Oct 25 2012 */

G.f.: 1/((1-t)*(1-t^2)*(1-t^3)*(1-t^4)*(1-t^5)*(1-t^6)*(1-t^7)*(1-t^8)). - N. J. A. Sloane, Jan 09 2016
a(n) = A008284(n+8, 8), n >= 0.
a(n) = floor((-1)^n*((n+1)*(-1)^(floor((n+2)/3)) + (2*n+3)*(-1)^(floor((n+1)/3)) + (n+2)*(-1)^(floor(n/3)))/972 + (n+2)*((-1)^n+1)*(-1)^(n/2)/512 + (n+18)*(6*n^6 + 648*n^5 + 27018*n^4 + 545616*n^3 + 5481213*n^2 + 25163028*n + 39226571)/1219276800 + (n+1)*(n^2+53*n+826)*(-1)^n/36864+1/2). (See link.) - Tani Akinari, Oct 26 2012
a(n) = a(n-1) + a(n-2) - a(n-5) - a(n-7) - a(n-9) + a(n-11) + 2*a(n-12) + a(n-13) + a(n-15) - a(n-16) - a(n-17) - 2*a(n-18) - a(n-19) - a(n-20) + a(n-21) + a(n-23) + 2*a(n-24) + a(n-25) - a(n-27) - a(n-29) - a(n-31) + a(n-34) + a(n-35) - a(n-36). - David Neil McGrath, Apr 14 2015
a(n+8) = a(n) + A008636(n). - Ece Uslu, Esin Becenen, Jan 11 2016
From Vladimír Modrák, Jul 30 2022: (Start)
a(n) = Sum_{i_1=0..floor(n/3)} Sum_{i_2=0..floor(n/4)} Sum_{i_3=0..floor(n/5)} Sum_{i_4=0..floor(n/6)} Sum_{i_5=0..floor(n/7)} Sum_{i_6=0..floor(n/8)} ceiling((max(0, n + 1 - 3*i_1 - 4*i_2 - 5*i_3 - 6*i_4 - 7*i_5 - 8*i_6))/2).
a(n) = Sum_{i_1=0..floor(n/4)} Sum_{i_2=0..floor(n/5)} Sum_{i_3=0..floor(n/6)} Sum_{i_4=0..floor(n/7)} Sum_{i_5=0..floor(n/8)} floor(((max(0, n + 3 - 4*i_1 - 5*i_2 - 6*i_3 - 7*i_4 - 8*i_5))^2+4)/12). (End)

More terms from Robert G. Wilson v, Dec 11 2000

A085756 Number of partitions into a prime number of distinct parts.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 20, 22, 26, 30, 35, 40, 47, 53, 62, 71, 82, 93, 108, 123, 141, 161, 184, 209, 239, 271, 309, 350, 397, 449, 509, 575, 649, 732, 825, 928, 1044, 1172, 1315, 1474, 1650, 1845, 2061, 2300, 2563, 2854, 3174, 3526, 3912, 4337

Offset: 3

Vladeta Jovovic, Jul 21 2003

nonn

			a(15)=20 because there are 20 partitions of 15 into a prime number of distinct parts: 1+2+3+4+5=4+5+6=3+5+7=2+6+7=3+4+8=2+5+8=1+6+8=7+8=2+4+9=1+5+9=6+9=2+3+10=
1+4+10=5+10=1+3+11=4+11=1+2+12=3+12=2+13=1+14.

Cf. A038499.

a(n) = A004526(n-1) +A001399(n-6) +A001401(n-15) +A008636(n-28) + .... - R. J. Mathar, Feb 13 2019

cp's OEIS Frontend

A026820 Euler's table: triangular array T read by rows, where T(n,k) = number of partitions in which every part is <= k for 1 <= k <= n. Also number of partitions of n into at most k parts.

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

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A008637 Number of partitions of n into at most 8 parts.

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A085756 Number of partitions into a prime number of distinct parts.

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A347543 Number of partitions of n into 7 or more parts.

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