cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A116861 Triangle read by rows: T(n,k) is the number of partitions of n such that the sum of the parts, counted without multiplicities, is equal to k (n>=1, k>=1).

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 1, 1, 2, 1, 0, 2, 1, 3, 1, 1, 3, 1, 1, 4, 1, 0, 3, 2, 2, 2, 5, 1, 1, 3, 3, 2, 4, 2, 6, 1, 0, 5, 2, 3, 4, 4, 3, 8, 1, 1, 4, 3, 4, 7, 4, 5, 3, 10, 1, 0, 5, 3, 4, 7, 7, 6, 6, 5, 12, 1, 1, 6, 4, 3, 12, 6, 8, 7, 9, 5, 15, 1, 0, 6, 4, 5, 10, 10, 9, 10, 11, 10, 7, 18, 1, 1, 6, 4, 5, 15, 11, 13, 9, 16, 11, 13, 8, 22
Offset: 1

Views

Author

Emeric Deutsch, Feb 27 2006

Keywords

Comments

Conjecture: Reverse the rows of the table to get an infinite lower-triangular matrix b with 1's on the main diagonal. The third diagonal of the inverse of b is minus A137719. - George Beck, Oct 26 2019
Proof: The reversed rows yield the matrix I+N where N is strictly lower triangular, N[i,j] = 0 for j >= i, having its 2nd diagonal equal to the 2nd column (1, 0, 1, 0, 1, ...): N[i+1,i] = A000035(i), i >= 1, and 3rd diagonal equal to the 3rd column of this triangle, (2, 1, 2, 3, 3, 3, ...): N[i+2,i] = A137719(i), i >= 1. It is known that (I+N)^-1 = 1 - N + N^2 - N^3 +- .... Here N^2 has not only the second but also the 3rd diagonal zero, because N^2[i+2,i] = N[i+2,i+1]*N[i+1,i] = A000035(i+1)*A000035(i) = 0. Therefore the 3rd diagonal of (I+N)^-1 is equal to -A137719 without leading 0. - M. F. Hasler, Oct 27 2019
From Gus Wiseman, Aug 27 2023: (Start)
Also the number of ways to write n-k as a nonnegative linear combination of a strict integer partition of k. Also the number of ways to write n as a (strictly) positive linear combination of a strict integer partition of k. Row n=7 counts the following:
7*1 . 1*2+5*1 1*3+4*1 1*3+2*2 1*5+2*1 1*7
2*2+3*1 2*3+1*1 1*4+3*1 1*3+1*2+2*1 1*4+1*3
3*2+1*1 1*5+1*2
1*6+1*1
1*4+1*2+1*1
(End)

Examples

			T(10,7) = 4 because we have [6,1,1,1,1], [4,3,3], [4,2,2,1,1] and [4,2,1,1,1,1] (6+1=4+3=4+2+1=7).
Triangle starts:
  1;
  1, 1;
  1, 0, 2;
  1, 1, 1, 2;
  1, 0, 2, 1, 3;
  1, 1, 3, 1, 1,  4;
  1, 0, 3, 2, 2,  2, 5;
  1, 1, 3, 3, 2,  4, 2, 6;
  1, 0, 5, 2, 3,  4, 4, 3, 8;
  1, 1, 4, 3, 4,  7, 4, 5, 3, 10;
  1, 0, 5, 3, 4,  7, 7, 6, 6,  5, 12;
  1, 1, 6, 4, 3, 12, 6, 8, 7,  9,  5, 15;
  ...

Links

Crossrefs

Cf. A000041 (row sums), A000009 (diagonal), A014153.
Cf. A114638 (count partitions with #parts = sum(distinct parts)).
Column 1: A000012, column 2: A000035(1..), column 3: A137719(1..).
For subsets instead of partitions we have A026820.
This statistic is ranked by A066328.
The central diagonal is T(2n,n) = A364910(n), non-strict A364907.
Partial sums of columns are columns of A364911.
Same as A364916 (offset 0) with rows reversed.
A008284 counts partitions by length, strict A008289.
A364912 counts linear combinations of partitions.
A364913 counts combination-full partitions, strict A364839.
Cf. A237667, A364272, A364915, A365002, A365004, A365006.

Programs

  • Maple
    g:= -1+product(1+t^j*x^j/(1-x^j), j=1..40): gser:= simplify(series(g,x=0,18)): for n from 1 to 14 do P[n]:=sort(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
# second Maple program:
b:= proc(n, i) option remember; local f, g, j;
      if n=0 then [1] elif i<1 then [ ] else f:= b(n, i-1);
         for j to n/i do
           f:= zip((x, y)->x+y, f, [0$i, b(n-i*j, i-1)[]], 0)
         od; f
      fi
    end:
T:= n-> subsop(1=NULL, b(n, n))[]:
seq(T(n), n=1..20);  # Alois P. Heinz, Feb 27 2013
  • Mathematica
    max = 14; s = Series[-1+Product[1+t^j*x^j/(1-x^j), {j, 1, max}], {x, 0, max}, {t, 0, max}] // Normal; t[n_, k_] := SeriesCoefficient[s, {x, 0, n}, {t, 0, k}]; Table[t[n, k], {n, 1, max}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 17 2014 *)
Table[Length[Select[IntegerPartitions[n],Total[Union[#]]==k&]],{n,0,10},{k,0,n}] (* Gus Wiseman, Aug 29 2023 *)
  • PARI
    A116861(n,k,s=0)={forpart(X=n,vecsum(Set(X))==k&&s++,k);s} \\ M. F. Hasler, Oct 27 2019

Formula

G.f.: -1 + Product_{j>=1} (1 + t^j*x^j/(1-x^j)).
Sum_{k=1..n} T(n,k) = A000041(n).
T(n,n) = A000009(n).
Sum_{k=1..n} k*T(n,k) = A014153(n-1).
T(n,1) = 1. T(n,2) = A000035(n+1). T(n,3) = A137719(n-2). - R. J. Mathar, Oct 27 2019
T(n,4) = A002264(n-1) + A121262(n). - R. J. Mathar, Oct 28 2019

A058398 Partition triangle A008284 read from right to left.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 3, 3, 1, 1, 1, 2, 3, 4, 3, 1, 1, 1, 2, 3, 5, 5, 4, 1, 1, 1, 2, 3, 5, 6, 7, 4, 1, 1, 1, 2, 3, 5, 7, 9, 8, 5, 1, 1, 1, 2, 3, 5, 7, 10, 11, 10, 5, 1, 1, 1, 2, 3, 5, 7, 11, 13, 15, 12, 6, 1, 1, 1, 2, 3, 5, 7, 11, 14, 18, 18, 14, 6, 1, 1, 1, 2, 3, 5, 7, 11
Offset: 1

Views

Author

Wolfdieter Lang, Dec 11 2000

Keywords

Comments

a(n,m) is the number of partitions of n with n-(m-1) parts or, equivalently, with greatest part n-(m-1).
The columns are the diagonals of triangle A008284. The diagonals are the columns of the partition array p(n,m), n >= 0, m >= 1, with p(n,m) the number of partitions of n in which every part is <= m; p(0,m) := 1. For n >= 1 this array is obtained from table A026820 read as lower triangular array with extension of the rows according to p(n,m)=A000041(n) for m>n.

Examples

			Lower triangular matrix:
  1;
  1,1;
  1,1,1;
  1,1,2,1;
  1,1,2,2,1;
  1,1,2,3,3,1;
  1,1,2,3,4,3,1;
  1,1,2,3,5,5,4,1;
  ...

References

  • L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 94, 96 and 307.
  • M. Kauers and P. Paule, The Concrete Tetrahedron, Springer 2011, p. 27.

Links

Crossrefs

Cf. A026820, A008284, A000041.

Programs

  • Mathematica
    row[n_] := Table[ IntegerPartitions[n, k] // Length, {k, 0, n}] // Differences // Reverse; Table[row[n], {n, 1, 14}] // Flatten (* Jean-François Alcover, Feb 28 2013 *)

Formula

a(n, m)= A008284(n, n-(m-1)).
a(n, m)= p(m-1, n-m+1), n >= m >= 1 with the p(n, m) array defined in the comment.
a(n, m)=0 if n
Viewed as a square array by antidiagonals, T(n,k) = 0 if n<0; T(n,1) = 1; otherwise T(n,k) = T(n,k-1) + T(n-k,k). - Franklin T. Adams-Watters, Jul 25 2006
Let x be a triangular number C(n,2), where n is the integer being partitioned. Then a(x) = a(x+1) = a(x+2) = 1. Also, a(x+3) = 2 for x>3 and a(x-1) = floor(n/2). - Allan Bickle, Apr 18 2024

A001400 Number of partitions of n into at most 4 parts.

Original entry on oeis.org

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, 1425, 1495
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Molien series for 4-dimensional representation of S_4 [Nebe, Rains, Sloane, Chap. 7].
Also number of pure 2-complexes on 4 nodes with n multiple 2-simplexes. - Vladeta Jovovic, Dec 27 1999
Also number of different integer triangles with perimeter <= n+3. Also number of different scalene integer triangles with perimeter <= n+9. - Reinhard Zumkeller, May 12 2002
a(n) is the coefficient of q^n in the expansion of (m choose 4)_q as m goes to infinity. - Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002
Also number of partitions of n into parts <= 4. a(n) = A026820(n,4), for n > 3. - Reinhard Zumkeller, Jan 21 2010
Number of different distributions of n+10 identical balls in 4 boxes as x,y,z,p where 0 < x < y < z < p. - Ece Uslu and Esin Becenen, Jan 11 2016
Number of partitions of 5n+8 or 5n+12 into 4 parts (+-) 3 mod 5. a(4) = 5 partitions of 28: [7,7,7,7], [12,7,7,2], [12,12,2,2], [17,7,2,2], [22,2,2,2]. a(3) = 3 partitions of 27: [8,8,8,3], [13,8,3,3], [18,3,3,3]. - Richard Turk, Feb 24 2016
a(n) is the total number of non-isomorphic geodetic graphs of diameter n homeomorphic to a complete graph K4. - Carlos Enrique Frasser, May 24 2018

Examples

			(4 choose 4)_q = 1, (5 choose 4)_q = q^4 + q^3 + q^2 + q + 1, (6 choose 4)_q = q^8 + q^7 + 2*q^6 + 2*q^5 + 3*q^4 + 2*q^3 + 2*q^2 + q + 1, (7 choose 4) = q^12 + q^11 + 2*q^10 + 3*q^9 + 4*q^8 + 4*q^7 + 5*q^6 + 4*q^5 + 4*q^4 + 3*q^3 + 2*q^2 + q + 1 so the coefficient of q^0 converges to 1, q^1 to 1, q^2 to 2 and so on.
G.f. = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 6*x^5 + 9*x^6 + 11*x^7 + ...
a(4) = 5, i.e., {1,2,3,8}, {1,2,4,7}, {1,2,5,6}, {2,3,4,5}, {1,3,4,6}. Number of different distributions of 14 identical balls in 4 boxes as x,y,z,p where 0 < x < y < z < p. - _Ece Uslu_, Esin Becenen, Jan 11 2016

References

  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 115, row m=4 of Q(m,n) table; p. 120, P(n,4).
  • H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 2.
  • 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, Addison-Wesley, 2005, Section 7.2.1.4., p. 56, exercise 31.
  • 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).

Links

Crossrefs

Essentially same as A026810. Partial sums of A005044.
a(n) = A008284(n+4, 4), n >= 0.
Cf. A008619, A001399, A001401, A026820, A070083, A117486.
First differences of A002621.

Programs

  • Haskell
    a001400 n = a001400_list !! n
a001400_list = scanl1 (+) a005044_list -- Reinhard Zumkeller, Feb 28 2013
  • Magma
    K:=Rationals(); M:=MatrixAlgebra(K,4); q1:=DiagonalMatrix(M,[1,-1,1,-1]); p1:=DiagonalMatrix(M,[1,1,-1,-1]); q2:=DiagonalMatrix(M,[1,1,1,-1]); h:=M![1,1,1,1, 1,1,-1,-1, 1,-1,1,-1, 1,-1,-1,1]/2; G:=MatrixGroup<4,K|q1,q2,h>; MolienSeries(G);
  • Maple
    A001400 := n->if n mod 2 = 0 then round(n^2*(n+3)/144); else round((n-1)^2*(n+5)/144); fi;
with(combstruct):ZL5:=[S,{S=Set(Cycle(Z,card<5))}, unlabeled]:seq(count(ZL5,size=n),n=0..55); # Zerinvary Lajos, Sep 24 2007
A001400:=-(-z**8+z**9+2*z**4-z**7-1-z)/(z**2+1)/(z**2+z+1)/(z+1)**2/(z-1)**4; # [conjectured by Simon Plouffe in his 1992 dissertation; gives sequence except for an initial 1]
B:=[S,{S = Set(Sequence(Z,1 <= card),card <=4)},unlabelled]: seq(combstruct[count](B, size=n), n=0..55); # Zerinvary Lajos, Mar 21 2009
  • Mathematica
    CoefficientList[ Series[ 1/((1 - x)*(1 - x^2)*(1 - x^3)*(1 - x^4)), {x, 0, 65} ], x ]
LinearRecurrence[{1, 1, 0, 0, -2, 0, 0, 1, 1, -1}, {1, 1, 2, 3, 5, 6, 9, 11, 15, 18}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 17 2012 *)
a[n_] := Sum[Floor[(n - j - 3*k + 2)/2], {j, 0, Floor[n/4]}, {k, j, Floor[(n - j)/3]}]; Table[a[n], {n, 0, 55}] (* L. Edson Jeffery, Jul 31 2014 *)
a[ n_] := With[{m = n + 5}, Round[ (2 m^3 - 3 m (5 + 3 (-1)^m)) / 288]]; (* Michael Somos, Dec 29 2014 *)
a[ n_] := With[{m = Abs[n + 5] - 5}, Sign[n + 5] Length[ IntegerPartitions[ m, 4]]]; (* Michael Somos, Dec 29 2014 *)
a[ n_] := With[{m = Abs[n + 5] - 5}, Sign[n + 5] SeriesCoefficient[ 1 / ((1 - x) (1 - x^2) (1 - x^3) (1 - x^4)), {x, 0, m}]]; (* Michael Somos, Dec 29 2014 *)
Table[Length@IntegerPartitions[n, 4], {n, 0, 55}] (* Robert Price, Aug 18 2020 *)
  • PARI
    a(n) = round(((n+4)^3 + 3*(n+4)^2 -9*(n+4)*((n+4)% 2))/144) \\ Washington Bomfim, Jul 03 2012
  • PARI
    {a(n) = n+=5; round( (2*n^3 - 3*n*(5 + 3*(-1)^n)) / 288)}; \\ Michael Somos, Dec 29 2014
  • PARI
    a(n) = #partitions(n,,4); \\ Ruud H.G. van Tol, Jun 02 2024

Formula

G.f.: 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)).
a(n) = 1 + (a(n-2) + a(n-3) + a(n-4)) - (a(n-5) + a(n-6) + a(n-7)) + a(n-9). - Norman J. Meluch (norm(AT)iss.gm.com), Mar 09 2000
P(n, 4) = (1/288)*(2*n^3 + 6*n^2 - 9*n - 13 + (9*n+9)*pcr{1, -1}(2, n) - 32*pcr{1, -1, 0}(3, n) - 36*pcr{1, 0, -1, 0}(4, n)) (see Comtet).
Let c(n) = Sum_{i=0..floor(n/3)} (1 + ceiling((n-3*i-1)/2)), then a(n) = Sum_{i=0..floor(n/4)} (1 + ceiling((n-4*i-1)/2) + c(n-4*i-3)). - Jon Perry, Jun 27 2003
Euler transform of finite sequence [1, 1, 1, 1].
(n choose 4)_q = (q^n-1)*(q^(n-1)-1)*(q^(n-2)-1)*(q^(n-3)-1)/((q^4-1)*(q^3-1)*(q^2-1)*(q-1)).
a(n) = round(((n+4)^3 + 3*(n+4)^2 - 9*(n+4)*((n+4) mod 2))/144). - Washington Bomfim, Jul 03 2012
a(n) = a(n-1) + a(n-2) - 2*a(n-5) + a(n-8) + a(n-9) - a(n-10). - David Neil McGrath, Sep 12 2014
a(n) = -a(-10-n) for all n in Z. - Michael Somos, Dec 29 2014
a(n) - a(n+1) - a(n+3) + a(n+4) = 0 if n is odd, else floor(n/4) + 2 for all n in Z. - Michael Somos, Dec 29 2014
a(n) = n^3/144 + n^2/24 - 7*n/144 + 1 + floor(n/4)/4 + floor(n/3)/3 + (n+5)*floor(n/2)/8 + floor((n+1)/4)/4. - Vaclav Kotesovec, Aug 18 2015
a(n) = a(n-4) + A001399(n). - Ece Uslu, Esin Becenen, Jan 11 2016, corrected Sep 25 2020
a(6*n) - a(6*n+1) - a(6*n+4) + a(6*n+5) = n+1. - Richard Turk, Apr 19 2016
a(n) = a(n-1) + A005044(n+3) for n>0, i.e., first differences is A005044. - Yuchun Ji, Oct 12 2020
From Vladimír Modrák and Zuzana Soltysova, Dec 09 2020: (Start)
a(n) = round((n + 3)^2/12) + Sum_{i=0..floor(n/4)} round((n - 4*i - 1)^2/12).
a(n) = floor(((n + 3)^2 + 4)/12) + Sum_{i=0..floor(n/4)} floor(((n - 4*i - 1)^2 + 4)/12). (End)
a(n) - a(n-3) = A008642(n). - R. J. Mathar, Jun 23 2021
a(n) - a(n-2) = A025767(n). - R. J. Mathar, Jun 23 2021
a(n) = round((2*n^3 + 30*n^2 + 135*n + 175)/288 + (-1)^n*(n+5)/32). - Dave Neary, Oct 28 2021
From Vladimír Modrák, Jul 13 2022: (Start)
a(n) = Sum_{j=0..floor(n/4)} Sum_{i=0..floor(n/3)} ceiling((max(0,n + 1 - 3*i - 4*j))/2).
a(n) = Sum_{i=0..floor(n/4)} floor(((n + 3 - 4*i)^2 + 4)/12). (End)
a(n) = floor(((n+4)^2*(n+7) - 9*(n+4)*(n mod 2) + 32)/144). - Vladimír Modrák, Mar 23 2025

A000065 -1 + number of partitions of n.

Original entry on oeis.org

0, 0, 1, 2, 4, 6, 10, 14, 21, 29, 41, 55, 76, 100, 134, 175, 230, 296, 384, 489, 626, 791, 1001, 1254, 1574, 1957, 2435, 3009, 3717, 4564, 5603, 6841, 8348, 10142, 12309, 14882, 17976, 21636, 26014, 31184, 37337, 44582, 53173, 63260, 75174, 89133, 105557, 124753
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

a(n+1) is the number of noncongruent n-dimensional integer-sided simplices with diameter n. - Sascha Kurz, Jul 26 2004
Also, the number of partitions of n into parts each less than n.
Also, the number of distinct types of equation which can be derived from the equation [n,0,0] not including itself. (Ince)
Also, the number of rooted trees on n+1 nodes with height exactly 2.
Also, the number of partitions (of any positive integer) whose sum + length is <= n. Example: a(5) = 6 counts 4, 3, 21, 2, 11, 1. Proof: Given a partition of n other than the all 1s partition, subtract 1 from each part and then drop the zeros. This is a bijection to the partitions with sum + length <= n. - David Callan, Nov 29 2007
Number of graphs with n vertices of treewidth n-2. Reason: The complement of a graph with n vertices and treewidth >= n-2 cannot have P3 or K3 as a subgraph (Chlebı́ková 2002, Theorem 10), so every component of it is a star. - Martín Muñoz, Dec 31 2023

Examples

			G.f. = x^2 + 2*x^3 + 4*x^4 + 6*x^5 + 10*x^6 + 14*x^7 + 21*x^8 + 29*x^9 + ...

References

  • E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944, p. 498; MR0010757.
  • 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).

Links

Crossrefs

A000041 - 1. A column of A058716. A diagonal of A263294.
Column h=2 of A034781.

Programs

  • Magma
    [NumberOfPartitions(n)-1: n in [0..50]]; // Vincenzo Librandi, Aug 25 2013
  • Maple
    with (combstruct):ZL:=proc(m) local i; [T0,{seq(T.i=Prod(Z,Set(T.(i+1))),i=0..m-1), T.m=Z}, unlabeled] end:A:=n -> count(ZL(2),size=n)-count(ZL(1),size=n): seq(A(n),n=1..46); # Zerinvary Lajos, Dec 05 2007
ZL :=[S, {S = Set(Cycle(Z),1 < card)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=0..45); # Zerinvary Lajos, Mar 25 2008
  • Mathematica
    nn=40;CoefficientList[Series[Product[1/(1-x^i),{i,1,nn}]-1/(1-x),{x,0,nn}],x]  (* Geoffrey Critzer, Oct 28 2012 *)
PartitionsP[Range[0,50]]-1 (* Harvey P. Dale, Aug 24 2013 *)
  • PARI
    {a(n) = if( n<0, 0, polcoeff( 1 / eta(x + x*O(x^n)), n) - 1)};
  • PARI
    {a(n) = if( n<0, 0, numbpart(n) - 1)};

Formula

a(n) = A026820(n,n-1) for n>1. - Reinhard Zumkeller, Jan 21 2010
G.f.: x*G(0)/(x-1) where G(k) = 1 - 1/(1-x^(k+1))/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 23 2013
G.f.: Sum_{k>=2} x^k / Product_{j=1..k} (1 - x^j). - Ilya Gutkovskiy, Sep 07 2021

A001401 Number of partitions of n into at most 5 parts.

Original entry on oeis.org

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, 4033, 4319
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

a(n) = T_{r}(n) for r large, where T_{r}(n) = number of outcomes in which r indistinguishable dice yield a sum r+n-1.
a(n) = coefficient of q^n in the expansion of (m choose 5)_q as m goes to infinity. - Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002
For n > 4: also number of partitions of n into parts <= 5: a(n) = A026820(n,5). - Reinhard Zumkeller, Jan 21 2010
Number of different distributions of n+15 identical balls in 5 boxes as x,y,z,p,q where 0 < x < y < z < p < q. - Ece Uslu and Esin Becenen, Jan 11 2016 [i.e., a(n) is the number of partitions of n+15 into 5 distinct parts. - R. J. Mathar, Feb 28 2021]
Tengely and Ulas prove that a(n) is a square only for n=1 and 2027. - Michel Marcus, Feb 11 2021

Examples

			(5 choose 5)_q = 1;
(6 choose 5)_q = q^5 + q^4 + q^3 + q^2 + q + 1;
(7 choose 5)_q = q^10 + q^9 + 2*q^8 + 2*q^7 + 3*q^6 + 3*q^5 + 3*q^4 + 2*q^3 + 2*q^2 + q + 1;
(8 choose 5)_q = q^15 + q^14 + 2*q^13 + 3*q^12 + 4*q^11 + 5*q^10 + 6*q^9 + 6*q^8 + 6*q^7 + 6*q^6 + 5*q^5 + 4*q^4 + 3*q^3 + 2*q^2 + q + 1;
so the coefficient of q^0 converges to 1, q^1 to 1, q^2 to 2 and so on.
a(3) = 3, i.e., {1,2,3,4,8}, {1,2,3,5,7}, {1,2,4,5,6}. Number of different distributions of 18 identical balls in 5 boxes as x,y,z,p,q where 0 < x < y < z < p < q. - _Ece Uslu_, Esin Becenen, Jan 11 2016

References

  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 115, row m=5 of Q(m,n) table.
  • H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 2.
  • 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.
  • 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).

Links

Crossrefs

a(n) = A008284(n+5, 5), n >= 0.
Cf. A008619, A001400, A001399, A008667 (first differences), A008804.
First differences of A002622.

Programs

  • Maple
    with(combstruct):ZL6:=[S,{S=Set(Cycle(Z,card<6))}, unlabeled]:seq(count(ZL6,size=n),n=0..52); # Zerinvary Lajos, Sep 24 2007
a:= n-> (Matrix(15, (i,j)-> if (i=j-1) then 1 elif j=1 then [1, 1, 0, 0, -1, -1, -1, 1, 1, 1, 0, 0, -1, -1, 1][i] else 0 fi)^n)[1,1]: seq(a(n), n=0..60); # Alois P. Heinz, Jul 31 2008
B:=[S,{S = Set(Sequence(Z,1 <= card),card <=5)},unlabelled]: seq(combstruct[count](B, size=n), n=0..52); # Zerinvary Lajos, Mar 21 2009
  • Mathematica
    CoefficientList[ Series[ 1/((1 - x)*(1 - x^2)*(1 - x^3)*(1 - x^4)*(1 - x^5)), {x, 0, 60} ], x ]
a[n_] := IntegerPartitions[n, 5] // Length; Table[a[n], {n, 0, 52}] (* Jean-François Alcover, Jul 13 2012 *)
LinearRecurrence[{1,1,0,0,-1,-1,-1,1,1,1,0,0,-1,-1,1},{1,1,2,3,5,7,10,13,18,23,30,37,47,57,70},60] (* Harvey P. Dale, Jan 05 2019 *)
  • PARI
    a(n)=#partitions(n,,5) \\ Charles R Greathouse IV, Sep 15 2014
  • PARI
    a(n) = (n^4 + 30*n^3 + 310*n^2 + 1320*n - 90*n*(n%2) + 2880)\2880 \\ Hoang Xuan Thanh, Aug 12 2025

Formula

G.f.: 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)).
a(n) = 1 + (a(n-2) + a(n-3) + a(n-4)) - (a(n-6) + (2*a(n-7)) + a(n-8)) + (a(n-10) + a(n-11) + a(n-12)) - a(n-14). - Norman J. Meluch (norm(AT)iss.gm.com), Mar 09 2000
Let a1(n) = Sum_{i=0..floor(n/3)} (1 + ceiling((n-3*i-1)/2)), a2(n) = Sum_{i=0..floor(n/4)} (1 + ceiling((n-4*i-1)/2) + a1(n-4*i-3)), then a(n) = Sum_{i=0..floor(n/5)} (1 + ceiling((n-5*i-1)/2) + a1(n-5*i-3) + a2(n-5*i-4)). - Jon Perry, Jun 27 2003
(n choose 5)_q=(q^n-1)*(q^(n-1)-1)*(q^(n-2)-1)*(q^(n-3)-1)*(q^(n-4)-1)/((q^5-1)*(q^4-1)*(q^3-1)*(q^2-1)*(q-1)).
a(n) = round(((n+5)^4 + 10*((n+5)^3 + (n+5)^2) - 75*(n+5) - 45*(n+5)*(-1)^(n+5))/2880). - Washington Bomfim, Jul 03 2012
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). - David Neil McGrath, Sep 13 2014
a(n+5) = a(n) + A001400(n) = A001400(n)+A026811(n). - Ece Uslu, Esin Becenen, Jan 11 2016
From Vladimír Modrák, Jul 13 2022: (Start)
a(n) = Sum_{k=0..floor(n/5)} Sum_{j=0..floor(n/4)} Sum_{i=0..floor(n/3)} ceiling((max(0, n + 1 - 3*i - 4*j - 5*k))/2).
a(n) = Sum_{j=0..floor(n/5)} Sum_{i=0..floor(n/4)} floor(((max(0, n + 3 - 4*i - 5*j))^2+4)/12). (End)
a(2n) = a(2n-1) + a(n) - a(n-8) = a(n) + Sum_{k=0..n-1} A008804(k). - David García Herrero, Aug 26 2024
a(n) = floor((n^4 + 30*n^3 + 310*n^2 + 1275*n + 45*n*(-1)^n+2880)/2880). - Hoang Xuan Thanh, Aug 12 2025

Extensions

Additional comments from Michael Somos and Branislav Kisacanin (branislav.kisacanin(AT)delphiauto.com)

A007042 Left diagonal of partition triangle A047812.

Original entry on oeis.org

0, 1, 3, 5, 9, 13, 20, 28, 40, 54, 75, 99, 133, 174, 229, 295, 383, 488, 625, 790, 1000, 1253, 1573, 1956, 2434, 3008, 3716, 4563, 5602, 6840, 8347, 10141, 12308, 14881, 17975, 21635, 26013, 31183, 37336, 44581, 53172, 63259, 75173, 89132, 105556, 124752
Offset: 1

Views

Author

N. J. A. Sloane, R. K. Guy

Keywords

Comments

For n > 2, a(n) is also the number of partitions of n into parts <= n-2: a(n) = A026820(n+1, n-1). - Reinhard Zumkeller, Jan 21 2010
Also, the number of partitions of 2*n in which n-1 is the maximal part; see the Mathematica section. - Clark Kimberling, Mar 13 2012
This is column 2 of the matrix A in Sect. 2.3 of the Govindarajan preprint, cf. references and A096651. - M. F. Hasler, Apr 12 2012

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000041, A007044, A007045, A026820, A047812, A051643, A096651.
Column k = 2 of A081719.

Programs

  • Julia
    using Nemo
function A007042List(len)
    R, z = PolynomialRing(ZZ, "z")
    e = eta_qexp(-1, len+2, z)
    [coeff(e, j) - 2 for j in 2:len+1] end
A007042List(45) |> println # Peter Luschny, May 30 2020
  • Mathematica
    f[n_]:= Length[Select[IntegerPartitions[2 n], First[#]==n-1 &]]; Table[f[n], {n, 1, 24}] (* Clark Kimberling, Mar 13 2012 *)
a[n_]:= PartitionsP[n+1]-2; Table[a[n], {n,1,50}] (* Jean-François Alcover, Jan 28 2015, after M. F. Hasler *)
  • PARI
    A007042(n)=numbpart(n+1)-2  \\ M. F. Hasler, Apr 12 2012

Formula

a(n) = A000041(n+1) - 2. - Vladeta Jovovic, Oct 06 2001

Extensions

More terms from James Sellers
Name edited by Petros Hadjicostas, May 31 2020

A001402 Number of partitions of n into at most 6 parts.

Original entry on oeis.org

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, 8442, 9192
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Also number of partitions of n into parts <= 6: a(n) = A026820(n,6). - Reinhard Zumkeller, Jan 21 2010
Counts unordered closed walks of weight n on a single vertex graph containing 6 loops of weights 1, 2, 3, 4, 5 and 6. - David Neil McGrath, Apr 11 2015
Number of different distributions of n+21 identical balls in 6 boxes as x,y,z,p,q,m where 0Ece Uslu and Esin Becenen, Jan 11 2016
a(n) could be the total number of non-isomorphic geodetic graphs of diameter n>=2 homeomorphic to the Petersen graph. - Carlos Enrique Frasser, May 24 2018

Examples

			The number of partitions of 6 into parts less than or equal to 6 is a(6)=11. These are (6)(51)(42)(33)(411)(321)(222)(3111)(2211)(21111)(111111). - _David Neil McGrath_, Apr 11 2015
a(4) = 5, i.e., {1,2,3,4,5,10},{1,2,3,4,6,9},{1,2,3,4,7,8},{1,2,3,5,6,8},{1,2,4,5,6,7} Number of different distributions of 25 identical balls in 6 boxes as x,y,z,p,q,m where 0 < x < y < z < p < q < m. - _Ece Uslu_, Esin Becenen, Jan 11 2016

References

  • A. Cayley, Calculation of the minimum N.G.F. of the binary seventhic, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 10, p. 408-419.
  • H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 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).

Links

Crossrefs

Essentially same as A026812. Cf. A037145 (first differences), A288341 (partial sums).
a(n) = A008284(n+6, 6), n >= 0.
A194197(n) = a(60*n). - Alois P. Heinz, Aug 23 2011

Programs

  • Maple
    with(combstruct):ZL7:=[S,{S=Set(Cycle(Z,card<7))}, unlabeled]: seq(count(ZL7,size=n),n=0..50);  # Zerinvary Lajos, Sep 24 2007
a:= n-> (Matrix(21, (i,j)-> if (i=j-1) then 1 elif j=1 then [1, 1, 0, 0, -1, 0, -2, 0, 1, 1, 1, 1, 0, -2, 0, -1, 0, 0, 1, 1, -1][i] else 0 fi)^n)[1,1]; seq(a(n), n=0..50);  # Alois P. Heinz, Jul 31 2008
B:=[S,{S = Set(Sequence(Z,1 <= card),card <=6)},unlabelled]: seq(combstruct[count](B, size=n), n=0..50); # Zerinvary Lajos, Mar 21 2009
## more efficient for large arguments (try with 10^100 or 100^1000):
a:= proc(n) local m, r; m := iquo (n, 60, 'r');
(167 +(2325 +(15400 +(47250 +54000*m +4500*r)*m +3150*r +150*r^2)*m
+[0, 795, 1875, 3030, 4500, 6075, 7995, 10050, 12480, 15075, 18075, 21270, 24900, 28755, 33075, 37650, 42720, 48075, 53955, 60150, 66900, 73995, 81675, 89730, 98400, 107475, 117195, 127350, 138180, 149475, 161475, 173970, 187200, 200955, 215475, 230550, 246420, 262875, 280155, 298050, 316800, 336195, 356475, 377430, 399300, 421875, 445395, 469650, 494880, 520875, 547875, 575670, 604500, 634155, 664875, 696450, 729120, 762675, 797355, 832950][r+1])*m
+[0, 63, 207, 348, 570, 795, 1143, 1482, 1968, 2475, 3135, 3828, 4722, 5643, 6795, 8010, 9468, 11007, 12843, 14760, 17010, 19383, 22107, 24978, 28260, 31695, 35583, 39672, 44238, 49035, 54375, 59958, 66132, 72603, 79695, 87120, 95238, 103707, 112923, 122550, 132960, 143823, 155547, 167748, 180870, 194535, 209163, 224382, 240648, 257535, 275535, 294228, 314082, 334683, 356535, 379170, 403128, 427947, 454143, 481260][r+1])*m/6
+[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, 8442, 9192, 9975, 10829, 11720, 12692, 13702, 14800, 15944, 17180, 18467][r+1] end:
seq(a(n), n=0..100);  # Alois P. Heinz, Aug 22 2011
A := [1,1,2,3,5,7,11,14,20,26,35,44,58,71,90,110,136,163,199,235,282];
a := proc(n) option remember; if n < 21 then A[n+1] else 1+(a(n-2)+a(n-3)+a(n-4))-(2*a(n-7)+2*a(n-8)+a(n-9))+(a(n-11)+2*a(n-12)+2*a(n-13))-(a(n-16)+a(n-17)+a(n-18))+(a(n-20)) fi end:
seq(a(i),i=0..50); # Peter Luschny, Aug 23 2011
## program using quasi-polynomials; see article by Sills and Zeilberger:
a:= m-> subs (n=m, add ([[n^5/86400 +7*n^4/11520 +77*n^3/6480 +245*n^2/2304 +43981*n/103680 +199577/345600], [-n^2/768 -7*n/256 -581/4608, n^2/768 +7*n/256 +581/4608], [-n/162 -19/324, -n/162 -23/324, n/81 +7/54], [1/32, -1/32, -1/32, 1/32], [1/25, 0, -1/25, -2/25, 2/25], [1/36, -1/36, -1/18, -1/36, 1/36, 1/18]][r][1 +irem (m-1+r, r)], r=1..6)):
seq(a(n), n=0..100);  # Alois P. Heinz, Aug 24 2011
## using Andrews-style expressions; see article by Sills and Zeilberger:
a:= n-> 1 +31*n^2/288 +floor(n/4)/16 -floor(n/4 +1/2)/16 +7*n^4/11520 +floor(n/5)/5 +n^5/86400 -(n^2/384 +7*n/128 +581/2304)*n +(n^2/192 +7*n/64 +581/1152) *floor(n/2) -(n/54 +61/324)*n +(n/54 +19/108) *floor((n+1)/3) +(n/27 +7/18) *floor(n/3) +floor(n/6)/18 -floor(n/6 +2/3)/36 +floor(n/6 +1/3)/18 +floor((n+1)/6)/12 +713*n/1800 +77*n^3/6480:
seq(a(n), n=0..100);  # Alois P. Heinz, Aug 24 2011
  • Mathematica
    CoefficientList[ Series[ 1/((1 - x)*(1 - x^2)*(1 - x^3)*(1 - x^4)*(1 - x^5)*(1 - x^6)), {x, 0, 60} ], x ]
(* 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]]]; a[n_] := T[n, 6]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 12 2017, after Alois P. Heinz's code for A026820 *)
Table[Length[IntegerPartitions[n,6]],{n,0,50}] (* Harvey P. Dale, Jul 30 2025 *)
  • PARI
    a(n)=floor((6*n^5+315*n^4+6160*n^3+55125*n^2+(216705+9600*(n%3<1))*n+527500)/518400+(n+1)*(n+20)*(-1)^n/768) \\ Tani Akinari, May 27 2014
  • PARI
    a(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))};
vector(60,n,n--; a(n)) \\ Washington Bomfim, Jan 16 2021

Formula

a(n) = 1 + (a(n-2) + a(n-3) + a(n-4)) - (2*a(n-7) + 2*a(n-8) + a(n-9)) + (a(n-11) + 2*a(n-12) + 2*a(n-13)) - (a(n-16) + a(n-17) + a(n-18)) + (a(n-20)). - Norman J. Meluch (norm(AT)iss.gm.com), Mar 09 2000
G.f.: 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)). - Alois P. Heinz, Aug 22 2011
a(n) ~ n^5 / 86400. - Charles R Greathouse IV, Aug 23 2011
a(n) = (167 + (2325 + (15400 + (47250 + 54000*m + 4500*r)*m + 3150*r + 150*r^2)*m + X(r))*m + Y(r))*m/6 + Z(r) where m = floor(n/60), r = n mod 60 and X, Y, Z are functions of r (see Maple program below). - Alois P. Heinz, Aug 23 2011
a(n) = floor((2 + 3*(floor(n/3) + floor(-n/3))) * (floor(n/3)+1)/54 + (6*n^5 + 315*n^4 + 6160*n^3 + 55125*n^2 + 219905*n + 485700)/518400 + (n+1)*(n+20)*(-1)^n/768). - Tani Akinari, Aug 05 2013
a(n) = a(n-1) + a(n-2) - a(n-5) - 2*a(n-7) + a(n-9) + a(n-10) + a(n-11) + a(n-12) - 2*a(n-14) - a(n-16) + a(n-19) + a(n-20) - a(n-21). - David Neil McGrath, Apr 11 2015
a(n+6) = a(n) + A001401(n). - Ece Uslu, Esin Becenen, Jan 11 2016
a(n) = round((n+11)*((6*n^4 + 249*n^3 + 2071*n^2 - 4931*n + 40621)/518400 + floor(n/2)*(n+10)/192 + (floor((n+1)/3) + 2*floor(n/3))/54)). - Washington Bomfim, Jan 15 2021

A322439 Number of ordered pairs of integer partitions of n where no part of the first is greater than any part of the second.

Original entry on oeis.org

1, 1, 3, 5, 11, 15, 33, 42, 82, 114, 195, 258, 466, 587, 954, 1317, 2021, 2637, 4124, 5298, 7995, 10565, 15075, 19665, 28798, 36773, 51509, 67501, 93060, 119299, 165589, 209967, 285535, 366488, 487536, 622509, 833998, 1048119, 1380410, 1754520, 2291406, 2876454
Offset: 0

Views

Author

Gus Wiseman, Dec 08 2018

Keywords

Examples

			The a(5) = 15 pairs of integer partitions:
      (5)|(5)
     (41)|(5)
     (32)|(5)
    (311)|(5)
    (221)|(5)
    (221)|(32)
   (2111)|(5)
   (2111)|(32)
  (11111)|(5)
  (11111)|(41)
  (11111)|(32)
  (11111)|(311)
  (11111)|(221)
  (11111)|(2111)
  (11111)|(11111)

Links

Crossrefs

Cf. A026794, A026820, A265947, A285573, A317144, A318915, A322435, A322436, A322440, A322441, A322442, A362051.

Programs

  • Maple
    g:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      g(n, i-1) +g(n-i, min(i, n-i)))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i>n, 0, b(n, i+1)+b(n-i, i)))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(g(n, i)*b(n-i, i), i=1..n))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Dec 09 2018
  • Mathematica
    Table[Length[Select[Tuples[IntegerPartitions[n],2],Max@@First[#]<=Min@@Last[#]&]],{n,20}]
(* Second program: *)
g[n_, i_] := g[n, i] = If[n == 0 || i == 1, 1, g[n, i - 1] + g[n - i, Min[i, n - i]]];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i>n, 0, b[n, i+1] + b[n-i, i]]];
a[n_] := a[n] = If[n == 0, 1, Sum[g[n, i]*b[n - i, i], {i, 1, n}]];
a /@ Range[0, 50] (* Jean-François Alcover, May 17 2021, after Alois P. Heinz *)

Formula

a(n) = Sum_{k = 1..n} A026820(n,k) * A026794(n,k).
a(n) = A000041(2n) - A362051(n) for n>=1. - Alois P. Heinz, Apr 27 2023

A173519 Number of partitions of n*(n+1)/2 into parts not greater than n.

Original entry on oeis.org

1, 1, 2, 7, 23, 84, 331, 1367, 5812, 25331, 112804, 511045, 2348042, 10919414, 51313463, 243332340, 1163105227, 5598774334, 27119990519, 132107355553, 646793104859, 3181256110699, 15712610146876, 77903855239751, 387609232487489, 1934788962992123
Offset: 0

Views

Author

Reinhard Zumkeller, Feb 20 2010

Keywords

Comments

a(n) is also the number of partitions of n^3 into n distinct parts <= n*(n+1). a(3) = 7: [4,11,12], [5,10,12], [6,9,12], [6,10,11], [7,8,12], [7,9,11], [8,9,10]. - Alois P. Heinz, Jan 25 2012

Links

Crossrefs

Cf. A066655, A097356, A258234.

Programs

  • Mathematica
    Table[Length[IntegerPartitions[n(n + 1)/2, n]], {n, 10}] (* Alonso del Arte, Aug 12 2011 *)
Table[SeriesCoefficient[Product[1/(1-x^k),{k,1,n}],{x,0,n*(n+1)/2}],{n,0,20}] (* Vaclav Kotesovec, May 25 2015 *)
  • PARI
    a(n)=
{
    local(tr=n*(n+1)/2, x='x+O('x^(tr+3)), gf);
    gf = 1 / prod(k=1,n, 1-x^k); /* g.f. for partitions into parts <=n */
    return( polcoeff( truncate(gf), tr ) );
} /* Joerg Arndt, Aug 14 2011 */

Formula

a(n) = A026820(A000217(n),n).
a(n) ~ c * d^n / n^2, where d = 5.4008719041181541524660911191042700520294... = A258234, c = 0.6326058791290010900659134913629203727... . - Vaclav Kotesovec, Sep 07 2014

Extensions

More terms from D. S. McNeil, Aug 12 2011

A008639 Number of partitions of n into at most 10 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
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

For n > 9: also number of partitions of n into parts <= 10: a(n) = A026820(n, 10). - Reinhard Zumkeller, Jan 21 2010

References

  • 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.

Links

Crossrefs

Essentially same as A026816.
a(n) = A008284(n + 10, 10), n >= 0.
Cf. A266778 (first differences), A288345 (partial sums).

Programs

  • Mathematica
    CoefficientList[ Series[ 1/ Product[ 1 - x^n, {n, 1, 10} ], {x, 0, 60} ], x ]
  • PARI
    Vec(1/prod(k=1,10,1-x^k)+O(x^99)) \\ Charles R Greathouse IV, May 06 2015

Formula

G.f.: 1/Product_{k=1..10} (1 - x^k). - David Neil McGrath, Apr 29 2015
a(n) = a(n-10) + A008638(n). - Vladimír Modrák, Sep 29 2020
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