A046746 -id:A046746 - cp's OEIS frontend

A028417 Sum over all n! permutations of n elements of minimum lengths of cycles.

1, 3, 10, 45, 236, 1505, 10914, 90601, 837304, 8610129, 96625970, 1184891081, 15665288484, 223149696601, 3394965018886, 55123430466945, 948479737691504, 17289345305870561, 332019600921360594, 6713316975465246889, 142321908843254560540, 3161718732648662557161

Offset: 1

Joe Keane (jgk(AT)jgk.org)

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..450

Cf. A028418, A046746, A006128.
Cf. A005225.
Column k=1 of A322383.

b:= proc(n, m) option remember; `if`(n=0, m, add((j-1)!*
      b(n-j, min(m,j))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, infinity):
seq(a(n), n=1..25);  # Alois P. Heinz, May 14 2016

Drop[Apply[Plus,Table[nn=25;Range[0,nn]!CoefficientList[Series[Exp[Sum[ x^i/i,{i,n,nn}]]-1,{x,0,nn}],x],{n,1,nn}]],1] (* Geoffrey Critzer, Jan 10 2013 *)
b[n_, m_] := b[n, m] = If[n == 0, m, Sum[(j-1)! b[n-j, Min[m, j]]* Binomial[n-1, j-1], {j, n}]];
a[n_] := b[n, Infinity];
Array[a, 25] (* Jean-François Alcover, Apr 21 2020, after Alois P. Heinz *)

E.g.f.: Sum[k>0, -1+ exp(Sum(j>=k, x^j/j))]. - Vladeta Jovovic, Jul 26 2004

a(n) = Sum_{k=1..n} k * A145877(n,k). - Alois P. Heinz, Jul 28 2014

More terms from Vladeta Jovovic, Sep 19 2002

A092265 Sum of smallest parts of all partitions of n into distinct parts.

Original entry on oeis.org

1, 2, 4, 5, 8, 10, 14, 16, 23, 26, 34, 40, 50, 58, 74, 83, 102, 120, 142, 164, 198, 226, 266, 308, 359, 412, 482, 548, 634, 730, 834, 950, 1094, 1240, 1416, 1609, 1826, 2068, 2350, 2648, 2994, 3382, 3806, 4280, 4826, 5408, 6070, 6806, 7619, 8522, 9534, 10632

Offset: 1

Vladeta Jovovic, Feb 14 2004

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A046746, A005895, A006128, A092269, A092319, A092316, A034296, A237665.

Cf. A026832, A336902, A336903.

b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i>n, 0, b(n,i+1)+b(n-i, i+1)))
    end:
a:= n-> add(j*b(n-j, j+1), j=1..n):
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 03 2016

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i > n, 0, b[n, i + 1] + b[n - i, i + 1]]]; a[n_] := Sum[j*b[n - j, j + 1], {j, 1, n}]; Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Jan 21 2017, after Alois P. Heinz *)

G.f.: Sum_{n >= 1} (-1 + Product_{k >= n} 1 + x^k).
G.f.: Sum_{n >= 1} n*x^n*Product_{k >= n+1} (1 + x^k). - Joerg Arndt, Jan 29 2011
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
Conjecture: a(n) = A034296(n) + A237665(n+1). - George Beck, May 06 2017
a(n) ~ exp(Pi*sqrt(n/3)) / (2 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, May 20 2018

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 25 2004

A097939 Sum of the smallest parts of all compositions of n.

Original entry on oeis.org

1, 3, 6, 12, 22, 42, 79, 151, 291, 566, 1106, 2175, 4293, 8499, 16864, 33523, 66727, 132958, 265137, 529050, 1056169, 2109282, 4213710, 8419697, 16827079, 33634489, 67237513, 134424624, 268768414, 537407062, 1074605619, 2148875961, 4297212424, 8593556211, 17185713097, 34369170909

Offset: 1

Vladeta Jovovic, Sep 05 2004

Sums of the antidiagonals of A099238. - Paul Barry, Oct 08 2004

Alois P. Heinz, Table of n, a(n) for n = 1..1000 (correcting an earlier b-file from Vincenzo Librandi)

Cf. A046746, A092309, A097940, A097941, A099238, A336902, A336903.

Column k=1 of A322427.

A097939:=n->add(add(binomial(n-r*(k+1)-1,k), k=0..floor((n-r-1)/(r+1))), r=0..n-1): seq(A097939(n), n=1..50); # Wesley Ivan Hurt, Dec 03 2016
# second Maple Program:
b:= proc(n, m) option remember; `if`(n=0, m,
      add(b(n-j, min(j, m)), j=1..n))
    end:
a:= n-> b(n$2):
seq(a(n), n=1..40);  # Alois P. Heinz, Jul 26 2020

Drop[ CoefficientList[ Series[ Sum[x^k/(1 - x - x^k), {k, 50}], {x, 0, 35}], x], 1] (* Robert G. Wilson v, Sep 08 2004 *)

N=66; x='x+O('x^N);
gf= sum(k=1,N, x^k/(1-x-x^k) );
Vec(gf)
/* Joerg Arndt, Jan 01 2013 */

{a(n)=polcoeff(sum(m=1,n,x^m*sumdiv(m,d,1/(1-x +x*O(x^n))^d) ),n)}

G.f.: Sum_{k>=1} x^k/(1-x-x^k).
a(n) = Sum_{r=0..n-1} Sum_{k=0..floor((n-r-1)/(r+1))} binomial(n-r(k+1)-1, k). - Paul Barry, Oct 08 2004
G.f.: (1-x)^2 * Sum_{k>=1} k*x^k/((x^k+x-1)*(x^(k+1)+x-1)). - Vladeta Jovovic, Apr 23 2006
G.f.: Sum_{k>=1} x^k/((1-x)^k*(1-x^k)). - Vladeta Jovovic, Mar 02 2008
G.f.: Sum_{n>=1} a*x^n/(1-a*x^n) (generalized Lambert series) where a=1/(1-x). - Joerg Arndt, Jan 30 2011
G.f.: Sum_{n>=1} (a*x)^n/(1-x^n) where a=1/(1-x). - Joerg Arndt, Jan 01 2013
G.f.: Sum_{n>=1} x^n * Sum_{d|n} 1/(1-x)^d. - Paul D. Hanna, Jul 18 2013
a(n) ~ 2^(n-1). - Vaclav Kotesovec, Oct 28 2014

More terms from Robert G. Wilson v, Sep 08 2004

A194437 Triangle read by rows: T(n,k) = sum of parts in the k-th region of n.

Original entry on oeis.org

1, 1, 3, 1, 3, 5, 1, 3, 5, 2, 9, 1, 3, 5, 2, 9, 3, 12, 1, 3, 5, 2, 9, 3, 12, 2, 6, 3, 20, 1, 3, 5, 2, 9, 3, 12, 2, 6, 3, 20, 3, 7, 4, 25, 1, 3, 5, 2, 9, 3, 12, 2, 6, 3, 20, 3, 7, 4, 25, 2, 6, 3, 13, 5, 4, 38, 1, 3, 5, 2, 9, 3, 12, 2, 6, 3, 20, 3, 7

Offset: 1

Omar E. Pol, Nov 27 2011

			Triangle begins:
1;
1,3;
1,3,5;
1,3,5,2,9;
1,3,5,2,9,3,12;
1,3,5,2,9,3,12,2,6,3,20;
1,3,5,2,9,3,12,2,6,3,20,3,7,4,25;
1,3,5,2,9,3,12,2,6,3,20,3,7,4,25,2,6,3,13,5,4,38;
...
Row n has length A000041(n). Row sums give A066186. Right border gives A046746. Records in every row give A046746. Rows converge to A186412.

Cf. A046746, A066186, A135010, A138121, A186114, A186412, A193870, A194436, A194438, A194439, A194446, A194447.

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

Original entry on oeis.org

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

A196931 Triangle read by rows in which row n lists in nondecreasing order the smallest part of every partition of n.

Original entry on oeis.org

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

Offset: 0

Omar E. Pol, Oct 21 2011

If n >= 1, row n lists the smallest parts of every partition of n in the order produced by the shell model of partitions of A135010, hence row n lists the parts of the last section of the set of partitions of n, except the emergent parts (See A182699).

Row n has length A000041(n). Row sums give A046746. Right border of triangle gives A001477. Row n starts with A000041(n-1) ones, n >= 1.

			Written as a triangle:
  0,
  1,
  1,2,
  1,1,3,
  1,1,1,2,4,
  1,1,1,1,1,2,5,
  1,1,1,1,1,1,1,2,2,3,6
  1,1,1,1,1,1,1,1,1,1,1,2,2,3,7,
  1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,4,8,
  ...

Mirror of triangle A182715.

Cf. A000041, A046746, A135010, A138121, A141285, A182699, A183152, A193827, A196930.

A116686 Total number of parts smaller than the largest part, in all partitions of n.

Original entry on oeis.org

0, 0, 1, 3, 8, 15, 29, 48, 79, 123, 188, 276, 404, 575, 808, 1122, 1540, 2089, 2811, 3748, 4958, 6519, 8504, 11034, 14231, 18268, 23312, 29638, 37486, 47245, 59279, 74140, 92347, 114703, 141933, 175174, 215478, 264407, 323448, 394788, 480509, 583609

Offset: 1

Emeric Deutsch, Feb 23 2006

nonn

Also, sum over all partitions of n of the difference between the largest part and the smallest part. - Franklin T. Adams-Watters, Feb 29 2008

Row sums of A244966. - Omar E. Pol, Jul 19 2014

			a(5) = 8 because the partitions of 5 are [5], [4,(1)], [3,(2)], [3,(1),(1)], [2,2,(1)], [2,(1),(1),(1)] and [1,1,1,1,1], containing a total of 8 parts that are smaller than the largest part (shown between parentheses).

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A116685, A211870, A211881.

f:=sum(x^i*sum(x^j/(1-x^j),j=1..i-1)/product(1-x^q,q=1..i),i=2..55): fser:=series(f,x=0,50): seq(coeff(fser,x^n),n=1..47);

Table[Length[Flatten[Rest[Split[#]]&/@Select[IntegerPartitions[n], #[[1]]> #[[-1]]&]]],{n,50}] (* Harvey P. Dale, Jul 26 2016 *)

a(n) = Sum_{k>=0} k*A116685(n,k).
G.f.: Sum_{i>=1} (x^i*(Sum_{j=1..i-1} x^j/(1-x^j))/(Product_{q=1..i} (1-x^q))).
a(n) = A006128(n) - A046746(n). - Vladeta Jovovic, Feb 24 2006
a(n) = A211870(n) + A211881(n). - Alois P. Heinz, Feb 13 2013

A210952 Triangle read by rows: T(n,k) = sum of all parts of the k-th column of the partitions of n but with the partitions aligned to the right margin.

Original entry on oeis.org

1, 1, 3, 1, 3, 5, 1, 3, 7, 9, 1, 3, 7, 12, 12, 1, 3, 7, 14, 21, 20, 1, 3, 7, 14, 24, 31, 25, 1, 3, 7, 14, 26, 40, 47, 38, 1, 3, 7, 14, 26, 43, 61, 66, 49, 1, 3, 7, 14, 26, 45, 70, 92, 93, 69, 1, 3, 7, 14, 26, 45, 73, 106, 130, 124, 87, 1, 3, 7, 14

Offset: 1

Omar E. Pol, Apr 22 2012

			For n = 6 the illustration shows the partitions of 6 aligned to the right margin and below the sums of the columns:
.
.                      6
.                  3 + 3
.                  4 + 2
.              2 + 2 + 2
.                  5 + 1
.              3 + 2 + 1
.              4 + 1 + 1
.          2 + 2 + 1 + 1
.          3 + 1 + 1 + 1
.      2 + 1 + 1 + 1 + 1
.  1 + 1 + 1 + 1 + 1 + 1
-------------------------
.  1,  3,  7, 14, 21, 20
.
So row 6 lists 1, 3, 7, 14, 21, 20.
Triangle begins:
1;
1, 3;
1, 3, 5;
1, 3, 7,  9;
1, 3, 7, 12, 12;
1, 3, 7, 14, 21, 20;
1, 3, 7, 14, 24, 31, 25;
1, 3, 7, 14, 26, 40, 47, 38;
1, 3, 7, 14, 26, 43, 61, 66, 49;
1, 3, 7, 14, 26, 45, 70, 92, 93, 69:

Mirror of triangle A206283. Rows sums give A066186. Rows converge to A014153. Right border gives A046746, >= 1.

Cf. A135010, A138121, A181187, A210763, A210950, A210951, A210953, A210961, A210970.

T(n,k) = Sum_{j=1..n} A210953(j,k). - Omar E. Pol, May 26 2012

A222044 Sum of smallest parts of all partitions of n into an odd number of parts.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 11, 15, 19, 28, 35, 47, 61, 80, 102, 136, 168, 218, 276, 350, 437, 556, 686, 860, 1063, 1321, 1620, 2005, 2443, 2998, 3649, 4445, 5377, 6531, 7863, 9496, 11398, 13694, 16373, 19603, 23347, 27834, 33058, 39259, 46467, 55020, 64914, 76599

Offset: 0

Alois P. Heinz, Feb 06 2013

nonn

a(n) + A222045(n) = A046746(n).

a(n) - A222045(n) = A222046(n).

			a(6) = 11: partitions of 6 into an odd number of parts are [2,1,1,1,1], [2,2,2], [3,2,1], [4,1,1], [6], sum of smallest parts is 1+2+1+1+6 = 11.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Cf. A046746, A211373, A222045, A222046, A222047, A222048, A222049.

b:= proc(n, i) option remember;
      [`if`(n=i, n, 0), 0]+`if`(i<1, [0, 0], b(n, i-1)+
       `if`(n [l[2], l[1]])(b(n-i, i))))
    end:
a:= n-> b(n, n)[1]:
seq(a(n), n=0..60);

b[n_, i_] := b[n, i] = {If[n==i, n, 0], 0} + If[i<1, {0, 0}, b[n, i-1] + If[nJean-François Alcover, Feb 03 2017, translated from Maple *)
Table[Total[Min/@Select[IntegerPartitions[n],OddQ[Length[#]]&]],{n,0,50}] (* Harvey P. Dale, Jul 05 2019 *)

a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*sqrt(3)*n). - Vaclav Kotesovec, Jul 06 2019

A222045 Sum of smallest parts of all partitions of n into an even number of parts.

Original entry on oeis.org

0, 0, 1, 1, 4, 4, 9, 10, 19, 21, 34, 40, 62, 72, 103, 124, 173, 207, 279, 337, 445, 538, 694, 842, 1077, 1299, 1634, 1977, 2464, 2969, 3669, 4411, 5410, 6488, 7896, 9447, 11442, 13640, 16421, 19536, 23411, 27761, 33124, 39174, 46554, 54915, 65008, 76485, 90258

Offset: 0

Alois P. Heinz, Feb 06 2013

nonn

A222044(n) + a(n) = A046746(n).

A222044(n) - a(n) = A222046(n).

			a(6) = 9: partitions of 6 into an even number of parts are [1,1,1,1,1,1], [2,2,1,1], [3,1,1,1], [3,3], [4,2], [5,1], sum of smallest parts is 1+1+1+3+2+1 = 9.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Cf. A046746, A211373, A222044, A222046, A222047, A222048, A222049.

b:= proc(n, i) option remember;
      [`if`(n=i, n, 0), 0]+`if`(i<1, [0, 0], b(n, i-1)+
       `if`(n [l[2], l[1]])(b(n-i, i))))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..60);

b[n_, i_] := b[n, i] = {If[n==i, n, 0], 0} + If[i<1, {0, 0}, b[n, i-1] + If[nJean-François Alcover, Feb 03 2017, translated from Maple *)

a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*sqrt(3)*n). - Vaclav Kotesovec, Jul 06 2019

cp's OEIS Frontend

A028417 Sum over all n! permutations of n elements of minimum lengths of cycles.

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A092265 Sum of smallest parts of all partitions of n into distinct parts.

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A097939 Sum of the smallest parts of all compositions of n.

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PARI

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A194437 Triangle read by rows: T(n,k) = sum of parts in the k-th region of n.

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

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A196931 Triangle read by rows in which row n lists in nondecreasing order the smallest part of every partition of n.

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A116686 Total number of parts smaller than the largest part, in all partitions of n.

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Maple

Mathematica

Formula

A210952 Triangle read by rows: T(n,k) = sum of all parts of the k-th column of the partitions of n but with the partitions aligned to the right margin.

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A222044 Sum of smallest parts of all partitions of n into an odd number of parts.

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