A026807 -id:A026807 - cp's OEIS frontend

A182715 Triangle read by rows in which row n lists in nonincreasing order the smallest part of every partition of n.

0, 1, 2, 1, 3, 1, 1, 4, 2, 1, 1, 1, 5, 2, 1, 1, 1, 1, 1, 6, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 7, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 4, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Omar E. Pol, Dec 01 2010

Triangle read by rows in which row n lists the smallest parts of all partitions of n in the order produced by the shell model of partitions of A138121.
Also, row n lists the "filler parts" of all partition of n. For more information see A182699.
Row n has length A000041(n). Row sums give A046746. Column 1 gives A001477. The last A000041(n-1) terms of row n are ones, n >= 1.

			For row 10, see the illustration of the link.
Triangle begins:
  0,
  1,
  2,1,
  3,1,1,
  4,2,1,1,1,
  5,2,1,1,1,1,1,
  6,3,2,2,1,1,1,1,1,1,1,
  7,3,2,2,1,1,1,1,1,1,1,1,1,1,1,
  8,4,3,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
  9,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
  ...

Omar E. Pol, Illustration of the smallest parts of all partitions of 10 (colored blue) in the shell model of partitions

Cf. A026807, A135010, A138121, A141285, A182699, A182709.

Mirror of triangle A196931.

Name simplified and more terms from Omar E. Pol, Oct 21 2011

A026835 Triangular array read by rows: T(n,k) = number of partitions of n into distinct parts in which every part is >=k, for k=1,2,...,n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 4, 2, 1, 1, 1, 1, 5, 3, 2, 1, 1, 1, 1, 6, 3, 2, 1, 1, 1, 1, 1, 8, 5, 3, 2, 1, 1, 1, 1, 1, 10, 5, 3, 2, 1, 1, 1, 1, 1, 1, 12, 7, 4, 3, 2, 1, 1, 1, 1, 1, 1, 15, 8, 5, 3, 2, 1, 1, 1, 1, 1, 1, 1, 18, 10, 6, 4, 3, 2, 1, 1, 1, 1, 1, 1, 1, 22, 12, 7, 4, 3, 2, 1, 1

Offset: 1

Clark Kimberling

T(n,1)=A000009(n), T(n,2)=A025147(n) for n>1, T(n,3)=A025148(n) for n>2, T(n,4)=A025149(n) for n>3.

A219922(n) = smallest number of row containing n. - Reinhard Zumkeller, Dec 01 2012

			From _Michael De Vlieger_, Aug 03 2020: (Start)
Table begins:
   1
   1   1
   2   1   1
   2   1   1   1
   3   2   1   1   1
   4   2   1   1   1   1
   5   3   2   1   1   1   1
   6   3   2   1   1   1   1   1
   8   5   3   2   1   1   1   1   1
  10   5   3   2   1   1   1   1   1   1
  12   7   4   3   2   1   1   1   1   1   1
  15   8   5   3   2   1   1   1   1   1   1   1
  ... (End)

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened

Cf. A026807.

Cf. A002260, A060016.

import Data.List (tails)
a026835 n k = a026835_tabl !! (n-1) !! (k-1)
a026835_row n = a026835_tabl !! (n-1)
a026835_tabl = map
   (\row -> map (p $ last row) $ init $ tails row) a002260_tabl
   where p 0      _ = 1
         p _     [] = 0
         p m (k:ks) = if m < k then 0 else p (m - k) ks + p m ks
-- Reinhard Zumkeller, Dec 01 2012

Nest[Function[{T, n, r}, Append[T, Table[1 + Total[T[[##]] & @@@ Select[r, #[[-1]] > k + 1 &]], {k, 0, n}]]] @@ {#1, #2, Transpose[1 + {#2 - #3, #3}]} & @@ {#1, #2, Range[Ceiling[#2/2] - 1]} & @@ {#, Length@ #} &, {{1}}, 12] // Flatten (* Michael De Vlieger, Aug 03 2020 *)

G.f.: Sum_{k>=1} (y^k*(-1+Product_{i>=k} (1+x^i))). - Vladeta Jovovic, Aug 25 2003
T(n, k) = 1 + Sum(T(i, j): i>=j>k and i+j=n+1). - Reinhard Zumkeller, Jan 01 2003
T(n, k) > 1 iff 2*k < n. - Reinhard Zumkeller, Jan 01 2003

A026832 Number of partitions of n into distinct parts, the least being odd.

Original entry on oeis.org

0, 1, 0, 2, 1, 2, 2, 4, 4, 5, 6, 8, 10, 12, 14, 18, 21, 24, 30, 36, 42, 50, 58, 68, 80, 93, 108, 126, 146, 168, 194, 224, 256, 294, 336, 384, 439, 500, 568, 646, 732, 828, 938, 1060, 1194, 1348, 1516, 1704, 1916, 2149, 2408, 2698, 3018, 3372, 3766, 4202, 4682

Offset: 0

Clark Kimberling

Fine's numbers L(n).

Also number of partitions of n such that if k is the largest part, then k occurs an odd number of times and each of the numbers 1,2,...,k-1 occurs at least once. Example: a(7)=4 because we have [3,2,1,1], [2,2,2,1], [2,1,1,1,1,1] and [1,1,1,1,1,1,1]. - Emeric Deutsch, Mar 29 2006

			a(7)=4 because we have [7], [6,1], [4,3] and [4,2,1].

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 56, Eq. (26.28).

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A026804, A026805, A026807, A026833, A092265, A096661, A097042.

Cf. A000009, A096765.

a026832 n = p 1 n where
   p _ 0 = 1
   p k m = if m < k then 0 else p (k+1) (m-k) + p (k+1+0^(n-m)) m
-- Reinhard Zumkeller, Jun 14 2012

g:=sum(x^(2*k-1)*product(1+x^j, j=2*k..60), k=1..60): gser:=series(g, x=0, 55): seq(coeff(gser, x, n), n=0..53); # Emeric Deutsch, Mar 29 2006
# second Maple program:
b:= proc(n, i) option remember; `if`(i*(i+1)/2 `if`(n=0, 0, b(n$2)):
seq(a(n), n=0..60);  # Alois P. Heinz, Feb 01 2019

mx=53; Rest[CoefficientList[Series[Sum[x^(2*k-1) Product[1+x^j, {j, 2*k, mx}], {k, mx}], {x, 0, mx}], x]]  (* Jean-François Alcover, Apr 05 2011, after Emeric Deutsch *)
Join[{0},Table[Length[Select[IntegerPartitions[n],OddQ[#[[-1]]]&&Max[Tally[#][[All,2]]] == 1&]],{n,60}]] (* Harvey P. Dale, May 14 2022 *)

G.f.: Sum_{k>=1} ((-1)^(k+1)*(-1+Product_{i>=k} (1+x^i))). - Vladeta Jovovic, Aug 26 2003
G.f.: Sum_{ k >= 1 } x^(k*(k+1)/2)/((1+x^k)*Product_{i=1..k} (1-x^i) ). - Vladeta Jovovic, Aug 10 2004
(1 + Sum_{n >= 1} a(n)q^n )*(1 + 2 Sum_{m>=1} (-1)^m*q^(m^2)) = Sum_{n >= 1} (-1)^n*q^((3*n^2+n)/2)/(1+q^n). [Fine]
G.f.: Sum_{k>=1} x^(2k-1)*Product_{j>=2k} (1 + x^j). - Emeric Deutsch, Mar 29 2006
a(n) ~ exp(Pi*sqrt(n/3)) / (2 * 3^(5/4) * n^(3/4)). - Vaclav Kotesovec, Jun 09 2019

More terms from Emeric Deutsch, Mar 29 2006

a(0)=0 prepended by Alois P. Heinz, Feb 01 2019

A299773 a(n) is the index of the partition that contains the divisors of n in the list of colexicographically ordered partitions of the sum of the divisors of n.

Original entry on oeis.org

1, 2, 3, 9, 7, 48, 15, 119, 72, 269, 56, 2740, 101, 1163, 1208, 5218, 297, 24319, 490, 42150, 6669, 14098, 1255, 792335, 5564, 42501, 30585, 432413, 4565, 4513067, 6842, 1251217, 122818, 317297, 124253, 54782479, 21637, 802541, 445414, 48590725, 44583

Offset: 1

Omar E. Pol, Mar 25 2018

nonn

If n is a noncomposite number (that is, 1 or prime), then a(n) = A000041(n).

For n >= 3, p(sigma(n-2)) < a(n) <= p(sigma(n-1)), where p(n) = A000041(n) and sigma(n) = A000203(n).

			For n = 4 the sum of the divisors of 4 is 1 + 2 + 4 = 7. Then we have that, in list of colexicographically ordered partitions of 7, the divisors of 4 are in the 9th partition, so a(4) = 9 (see below):
------------------------------------------------------
   k        Diagram        Partitions of 7
------------------------------------------------------
         _ _ _ _ _ _ _
   1    |_| | | | | | |    [1, 1, 1, 1, 1, 1, 1]
   2    |_ _| | | | | |    [2, 1, 1, 1, 1, 1]
   3    |_ _ _| | | | |    [3, 1, 1, 1, 1]
   4    |_ _|   | | | |    [2, 2, 1, 1, 1]
   5    |_ _ _ _| | | |    [4, 1, 1, 1]
   6    |_ _ _|   | | |    [3, 2, 1, 1]
   7    |_ _ _ _ _| | |    [5, 1, 1]
   8    |_ _|   |   | |    [2, 2, 2, 1]
   9    |_ _ _ _|   | |    [4, 2, 1]       <---- Divisors of 4
  10    |_ _ _|     | |    [3, 3, 1]
  11    |_ _ _ _ _ _| |    [6, 1]
  12    |_ _ _|   |   |    [3, 2, 2]
  13    |_ _ _ _ _|   |    [5, 2]
  14    |_ _ _ _|     |    [4, 3]
  15    |_ _ _ _ _ _ _|    [7]
.

Amiram Eldar, Table of n, a(n) for n = 1..3200 (terms 1..700 from Andrew Howroyd)

Cf. A000040, A000041, A000203, A008578, A026807, A027750, A056538, A135010, A141285, A194446, A211992, A272024.

b[n_, k_] := b[n, k] = If[k < 1 || k > n, 0, If[n == k, 1, b[n, k + 1] + b[n - k, k]]];
PartIndex[v_] := Module[{s = 1, t = 0}, For[i = Length[v], i >= 1, i--, t += v[[i]]; s += b[t, If[i == 1, 1, v[[i - 1]]]] - b[t, v[[i]]]]; s];
a[n_] := PartIndex[Divisors[n]];
a /@ Range[1, 100] (* Jean-François Alcover, Sep 27 2019, after Andrew Howroyd *)

a(n)={my(d=divisors(n)); vecsearch(vecsort(partitions(vecsum(d))), d)} \\ Andrew Howroyd, Jul 15 2018

\\ here b(n,k) is A026807.
b(n,k)=polcoeff(1/prod(i=k, n, 1-x^i + O(x*x^n)), n)
PartIndex(v)={my(s=1,t=0); forstep(i=#v, 1, -1, t+=v[i]; s+=b(t, if(i==1, 1, v[i-1])) - b(t, v[i])); s}
a(n)=PartIndex(divisors(n)); \\ Andrew Howroyd, Jul 15 2018

Terms a(8) and beyond from Andrew Howroyd, Jul 15 2018

A057623 a(n) = n! * (sum of reciprocals of all parts in unrestricted partitions of n).

Original entry on oeis.org

1, 5, 29, 218, 1814, 18144, 196356, 2427312, 32304240, 475637760, 7460546400, 127525829760, 2302819079040, 44659367020800, 911770840108800, 19784985947596800, 449672462639769600, 10790180876185804800, 270071861749240320000, 7094011359005190144000

Offset: 1

Leroy Quet, Oct 09 2000

nonn

			The unrestricted partitions of 3 are 1 + 1 + 1, 1 + 2 and 3. So a(3) = 3! *(1 + 1 + 1 + 1 + 1/2 + 1/3) = 29.

Alois P. Heinz, Table of n, a(n) for n = 1..400
Guo-Niu Han, An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths, arXiv:0804.1849 [math.CO]; see p.27

Column 1 of A210590.

Cf. A103738.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
       b(n, i-1)+`if`(i>n, 0, (p-> p+[0, p[1]/i])(b(n-i, i)))))
    end:
a:= n-> n!*b(n$2)[2]:
seq(a(n), n=1..30);  # Alois P. Heinz, Sep 11 2014

b[n_, i_] := b[n, i] = If[n==0, {1, 0}, If[i<1, 0, b[n, i-1] + If[i>n, 0, Function[ {p}, p + {0, p[[1]]/i}][b[n-i, i]]]]]; a[n_] := n!*b[n, n][[2]]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Apr 02 2015, after Alois P. Heinz *)
Table[n!*Sum[DivisorSigma[1, k]*PartitionsP[n - k]/k, {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, May 29 2018 *)

S(n,m):=(if n=m then 1 else if nVladimir Kruchinin, Sep 10 2014 */

{a(n) = my(t='t); n!*polcoef(polcoef(prod(k=1, n, (1-x^k+x*O(x^n))^(-1-t)), n), 1)} \\ Seiichi Manyama, Nov 07 2020

n! *sum_{k=1 to n} [sigma(k) p(n-k) /k], where sigma(n) = sum of positive divisors of n and p(n) = number of unrestricted partitions of n.

a(n) = P(n,1), where P(n,m) = P(n,m+1)+S(n-m,m)*n!/m+n!/(n-m)!*P(n-m,m)), P(n,n)=(n-1)!, P(n,m)=0 for m>n, S(n,m) is triangle of A026807. - Vladimir Kruchinin, Sep 10 2014

A027199 Triangular array T read by rows: T(n,k) = number of partitions of n into an odd number of parts, each >=k.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 1, 5, 2, 1, 1, 1, 1, 8, 2, 1, 1, 1, 1, 1, 10, 3, 1, 1, 1, 1, 1, 1, 16, 4, 2, 1, 1, 1, 1, 1, 1, 20, 6, 2, 1, 1, 1, 1, 1, 1, 1, 29, 7, 3, 1, 1, 1, 1, 1, 1, 1, 1, 37, 10, 4, 2, 1, 1, 1, 1, 1, 1, 1, 1, 52, 12, 5, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 66, 17, 6, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Clark Kimberling

			Triangle begins:
   1;
   1,  1;
   2,  1, 1;
   2,  1, 1, 1;
   4,  1, 1, 1, 1;
   5,  2, 1, 1, 1, 1;
   8,  2, 1, 1, 1, 1, 1;
  10,  3, 1, 1, 1, 1, 1, 1;
  16,  4, 2, 1, 1, 1, 1, 1, 1;
  20,  6, 2, 1, 1, 1, 1, 1, 1, 1;
  29,  7, 3, 1, 1, 1, 1, 1, 1, 1, 1;
  37, 10, 4, 2, 1, 1, 1, 1, 1, 1, 1, 1;
  52, 12, 5, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1;

Cf. A027185, A027193 (first column), A027194, A027195, A027196, A027197, A027198.

T(n, k) = polcoef(x^k*sum(i=0, n, x^(2*k*i)/prod(j=1, 2*i+1, 1-x^j+x*O(x^n))), n); \\ Seiichi Manyama, May 15 2023

 T(n, k) = Sum{O(n, i)}, k<=i<=n, O given by A027185.
T(n,k) + A027200(n,k) = A026807(n,k). - R. J. Mathar, Oct 18 2019
G.f. of column k: x^k * Sum_{i>=0} x^(2*k*i)/Product_{j=1..2*i+1} (1-x^j). - Seiichi Manyama, May 15 2023

More terms from Seiichi Manyama, May 15 2023

A027200 Triangular array T read by rows: T(n,k) = number of partitions of n into an even number of parts, each >=k.

Original entry on oeis.org

0, 1, 0, 1, 0, 0, 3, 1, 0, 0, 3, 1, 0, 0, 0, 6, 2, 1, 0, 0, 0, 7, 2, 1, 0, 0, 0, 0, 12, 4, 2, 1, 0, 0, 0, 0, 14, 4, 2, 1, 0, 0, 0, 0, 0, 22, 6, 3, 2, 1, 0, 0, 0, 0, 0, 27, 7, 3, 2, 1, 0, 0, 0, 0, 0, 0, 40, 11, 5, 3, 2, 1, 0, 0, 0, 0, 0, 0, 49, 12, 5, 3, 2, 1, 0, 0, 0, 0, 0, 0, 0, 69, 17, 7, 4, 3, 2, 1, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Clark Kimberling

			 Triangle begins:
   0;
   1,  0;
   1,  0, 0;
   3,  1, 0, 0;
   3,  1, 0, 0, 0;
   6,  2, 1, 0, 0, 0;
   7,  2, 1, 0, 0, 0, 0;
  12,  4, 2, 1, 0, 0, 0, 0;
  14,  4, 2, 1, 0, 0, 0, 0, 0;
  22,  6, 3, 2, 1, 0, 0, 0, 0, 0;
  27,  7, 3, 2, 1, 0, 0, 0, 0, 0, 0;
  40, 11, 5, 3, 2, 1, 0, 0, 0, 0, 0, 0;
  49, 12, 5, 3, 2, 1, 0, 0, 0, 0, 0, 0, 0;

Cf. A027186, A027187 (1st column), A027188, A027189, A027190, A027191, A027192.

T(n, k) = polcoef(sum(i=0, n, x^(2*k*i)/prod(j=1, 2*i, 1-x^j+x*O(x^n))), n); \\ Seiichi Manyama, May 15 2023

 T(n, k) = Sum{E(n, i)}, k<=i<=n, E given by A027186.
T(n,k) + A027199(n,k) = A026807(n,k). - R. J. Mathar, Oct 18 2019
G.f. of column k: Sum_{i>=0} x^(2*k*i)/Product_{j=1..2*i} (1-x^j). - Seiichi Manyama, May 15 2023

More terms from Seiichi Manyama, May 15 2023

cp's OEIS Frontend

A182715 Triangle read by rows in which row n lists in nonincreasing order the smallest part of every partition of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A026835 Triangular array read by rows: T(n,k) = number of partitions of n into distinct parts in which every part is >=k, for k=1,2,...,n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

A026832 Number of partitions of n into distinct parts, the least being odd.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Formula

Extensions

A299773 a(n) is the index of the partition that contains the divisors of n in the list of colexicographically ordered partitions of the sum of the divisors of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Extensions

A057623 a(n) = n! * (sum of reciprocals of all parts in unrestricted partitions of n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

PARI

Formula

A027199 Triangular array T read by rows: T(n,k) = number of partitions of n into an odd number of parts, each >=k.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

Extensions

A027200 Triangular array T read by rows: T(n,k) = number of partitions of n into an even number of parts, each >=k.

Views

Author

Keywords

Examples

Crossrefs

Programs