A008639 -id:A008639 - 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

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

A288345 Expansion of 1 / ((1-x)^2(1-x^2)(1-x^3)...(1-x^10)).

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 45, 67, 97, 139, 194, 269, 366, 494, 658, 870, 1137, 1477, 1900, 2430, 3083, 3890, 4874, 6078, 7533, 9294, 11406, 13940, 16955, 20545, 24787, 29800, 35688, 42600, 50670, 60088, 71024, 83714, 98377, 115305, 134771, 157138, 182746, 212038

Offset: 0

Seiichi Manyama, Jun 08 2017

Number of partitions of at most n into at most 10 parts.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Richard J. Mathar, Size of the Set of Residues of Integer Powers of Fixed Exponent, research paper, 2017.
Index entries for linear recurrences with constant coefficients, signature (2, 0, -1, 0, -1, 1, -1, 1, 0, 0, -1, 2, 0, 0, 1, -2, 0, -1, 0, 0, 0, -2, 3, 0, 1, 0, 1, 0, -1, 0, -1, 0, -3, 2, 0, 0, 0, 1, 0, 2, -1, 0, 0, -2, 1, 0, 0, -1, 1, -1, 1, 0, 1, 0, -2, 1).

Number of partitions of at most n into at most k parts: A002621 (k=4), A002622 (k=5), A288341 (k=6), A288342 (k=7), A288343 (k=8), A288344 (k=9), this sequence (k=10).

Cf. A008639 (first differences).

x='x+O('x^99); Vec(1/((1-x)*prod(i=1, 10, (1-x^i)))) \\ Altug Alkan, Mar 28 2018

A341914 Number of partitions of n into 10 distinct and relatively prime parts.

Original entry on oeis.org

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, 22367, 25608, 29292, 33401, 38047, 43214, 49037, 55494, 62740, 70760, 79725, 89623

Offset: 55

Ilya Gutkovskiy, Feb 23 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 55..1000

Cf. A023022, A023030, A023035, A078374, A101271, A339672, A340719, A341868, A341870, A341912, A341913.

nmax = 108; CoefficientList[Series[Sum[MoebiusMu[k] x^(55 k)/Product[1 - x^(j k), {j, 1, 10}], {k, 1, nmax}], {x, 0, nmax}], x] // Drop[#, 55] &

G.f.: Sum_{k>=1} mu(k)* x^(55*k) / Product_{j=1..10} (1 - x^(j*k)).

a(n) <= A008639(n-55), equality for n<110. - R. J. Mathar, Feb 28 2021

A347547 Number of partitions of n into 10 or more parts.

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 45, 67, 97, 139, 194, 270, 368, 499, 667, 887, 1165, 1524, 1973, 2544, 3253, 4143, 5239, 6602, 8268, 10320, 12813, 15859, 19537, 24000, 29359, 35820, 43541, 52795, 63803, 76929, 92476, 110926, 132694, 158414, 188649, 224231, 265916, 314793

Offset: 10

Ilya Gutkovskiy, Sep 06 2021

nonn

Cf. A000065, A004250, A008639, A035300, A035301, A347542, A347543, A347544, A347545.

nmax = 54; CoefficientList[Series[Sum[x^k/Product[(1 - x^j), {j, 1, k}], {k, 10, nmax}], {x, 0, nmax}], x] // Drop[#, 10] &

G.f.: Sum_{k>=10} x^k / Product_{j=1..k} (1 - x^j).

A008633 Molien series for A_10.

Original entry on oeis.org

Offset: 0

N. J. A. Sloane

D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 105.

Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (0, 2, 2, 0, -2, -3, -3, 0, 3, 3, 0, 0, 1, 2, 3, 1, -5, -8, -5, 0, 4, 4, 2, 0, 2, 4, 4, 0, -5, -8, -5, 1, 3, 2, 1, 0, 0, 3, 3, 0, -3, -3, -2, 0, 2, 2, 0, -1).

(1+x^45)/(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)

G.f. (1+x^45)/ prod_{m=1..10} (1-x^m).

a(n) = A008639(n) + A008639(n-45). - R. J. Mathar, Mar 12 2023

More terms from R. J. Mathar, May 24 2008

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

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A288345 Expansion of 1 / ((1-x)^2*(1-x^2)*(1-x^3)*...*(1-x^10)).

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A341914 Number of partitions of n into 10 distinct and relatively prime parts.

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A347547 Number of partitions of n into 10 or more parts.

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A008633 Molien series for A_10.

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A288345 Expansion of 1 / ((1-x)^2(1-x^2)(1-x^3)...(1-x^10)).