A046746 -id:A046746 - cp's OEIS frontend

A336902 Sum of the smallest parts of all compositions of n into distinct parts.

0, 1, 2, 5, 6, 11, 18, 25, 32, 53, 84, 107, 156, 205, 302, 497, 618, 863, 1206, 1597, 2228, 3569, 4440, 6191, 8256, 11329, 14642, 20477, 30390, 38555, 52578, 69625, 92696, 122141, 160500, 211955, 310476, 386941, 521102, 678617, 901386, 1155383, 1529742, 1940749

Offset: 0

Alois P. Heinz, Aug 07 2020

nonn

			a(6) = 18 = 1 + 1 + 1 + 1 + 1 + 1 + 2 + 2 + 1 + 1 + 6: (1)23, (1)32, 2(1)3, 23(1), 3(1)2, 32(1), (2)4, 4(2), (1)5, 5(1), (6).

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Cf. A005895, A006128, A046746, A092265, A097939, A102712, A336903.

b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 b(n$2, 1):
seq(a(n), n=0..50);

b[n_, i_, p_] := b[n, i, p] = If[i(i+1)/2 < n || i < 1, 0,
     If[i == n, i*p!, b[n-i, Min[n-i, i-1], p+1]] + b[n, i-1, p]];
a[n_] := b[n, n, 1];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 12 2021, after Alois P. Heinz *)

a(n) == n (mod 2).

A336903 Sum of the largest parts of all compositions of n into distinct parts.

Original entry on oeis.org

0, 1, 2, 7, 10, 19, 42, 61, 98, 151, 304, 403, 654, 925, 1400, 2431, 3328, 4903, 7056, 10117, 13952, 23419, 30406, 44683, 61308, 87289, 116822, 164359, 247774, 327715, 457542, 624445, 855062, 1148023, 1559188, 2058643, 3043506, 3906637, 5375732, 7111975, 9679852

Offset: 0

Alois P. Heinz, Aug 07 2020

nonn

			a(6) = 42 = 3 + 3 + 3 + 3 + 3 + 3 + 4 + 4 + 5 + 5 + 6: 12(3), 1(3)2, 21(3), 2(3)1, (3)12, (3)21, 2(4), (4)2, 1(5), (5)1, (6).

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Cf. A005895, A006128, A046746, A092265, A097939, A102712, A336902.

b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 `if`(n=0, 0, b(n$2, 0)):
seq(a(n), n=0..50);

b[n_, i_, p_] := b[n, i, p] = If[i(i + 1)/2 < n, 0,
     If[n == 0, p!, b[n - i, Min[n - i, i - 1], p + 1]*
     If[p == 0, i, 1] + b[n, i - 1, p]]];
a[n_] := If[n == 0, 0, b[n, n, 0]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 12 2021, after Alois P. Heinz *)

a(n) == n (mod 2).

A097147 Total sum of minimum block sizes in all partitions of n-set.

Original entry on oeis.org

1, 3, 7, 21, 66, 258, 1079, 4987, 25195, 136723, 789438, 4863268, 31693715, 217331845, 1564583770, 11795630861, 92833623206, 760811482322, 6479991883525, 57256139503047, 523919025038279, 4956976879724565, 48424420955966635, 487810283307069696

Offset: 1

Vladeta Jovovic, Jul 27 2004

Alois P. Heinz, Table of n, a(n) for n = 1..576

Cf. A028417, A028418, A046746, A006128, A097145, A097146, A097148.

Column k=1 of A319298.

g:= proc(n, i, p) option remember; `if`(n=0, (i+1)*p!,
      `if`(i<1, 0, add(g(n-i*j, i-1, p+j*i)/j!/i!^j, j=0..n/i)))
    end:
a:= n-> g(n$2, 0):
seq(a(n), n=1..30);  # Alois P. Heinz, Mar 06 2015

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 *)
g[n_, i_, p_] := g[n, i, p] = If[n == 0, (i+1)*p!, If[i<1, 0,
     Sum[g[n-i*j, i-1, p+j*i]/j!/i!^j, {j, 0, n/i}]]];
a[n_] := g[n, n, 0];
Array[a, 30] (* Jean-François Alcover, Aug 24 2021, after Alois P. Heinz *)

E.g.f.: Sum_{k>0} (-1+exp(Sum_{j>=k} x^j/j!)).

More terms from Max Alekseyev, Apr 29 2010

A097148 Total sum of maximum block sizes in all partitions of n-set.

Original entry on oeis.org

1, 3, 10, 35, 136, 577, 2682, 13435, 72310, 414761, 2524666, 16239115, 109976478, 781672543, 5814797281, 45155050875, 365223239372, 3070422740989, 26780417126048, 241927307839731, 2260138776632752, 21805163768404127, 216970086170175575, 2224040977932468379

Offset: 1

Vladeta Jovovic, Jul 27 2004

Let M be the infinite lower triangular matrix given by A080510 and v the column vector [1,2,3,...] then M*v=A097148 (this sequence, as column vector). - Gary W. Adamson, Feb 24 2011

Alois P. Heinz, Table of n, a(n) for n = 1..576

Cf. A028417, A028418, A046746, A006128, A097145-A097147.
Cf. A080510.
Column k=1 of A319375.

b:= proc(n, m) option remember; `if`(n=0, m, add(
      b(n-j, max(j, m))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=1..24);  # Alois P. Heinz, Mar 02 2020, revised May 08 2024

Drop[ Range[0, 22]! CoefficientList[ Series[ Sum[E^(E^x - 1) - E^Sum[x^j/j!, {j, 1, k}], {k, 0, 22}], {x, 0, 22}], x], 1] (* Robert G. Wilson v, Aug 05 2004 *)

E.g.f.: Sum_{k>=0} (exp(exp(x)-1)-exp(Sum_{j=1..k} x^j/j!)).

More terms from Robert G. Wilson v, Aug 05 2004

A182707 Sum of the parts of all partitions of n-1 plus the sum of the emergent parts of the partitions of n.

Original entry on oeis.org

0, 1, 4, 11, 23, 46, 80, 138, 221, 351, 529, 801, 1161, 1685, 2380, 3355, 4624, 6375, 8623, 11658, 15538, 20664, 27163, 35660, 46330, 60082, 77288, 99197, 126418, 160802, 203246, 256381, 321700, 402781, 501962, 624332, 773235, 955776, 1177076, 1446762, 1772308

Offset: 1

Omar E. Pol, Nov 28 2010

For more information about the emergent parts of the partitions of n see A182699 and A182709.

			For n = 6 the partitions of 6-1=5 are (5);(3+2);(4+1);(2+2+1);(3+1+1);(2+1+1+1);(1+1+1+1+1) and the sum of the parts give 35, the same as 5*7. By other hand the emergent parts of the partitions of 6 are (2+2);(4);(3) and the sum give 11, so a(6) = 35+11 = 46.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Jason Kimberley)

Cf. A000041, A046746, A066186, A135010, A138121, A182699, A182708, A182709.

a(n) = A066186(n) - A046746(n) = A066186(n-1) + A182709(n).

a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)) * (1 - (sqrt(3/2)/Pi + Pi/(24*sqrt(6)))/sqrt(n)). - Vaclav Kotesovec, Jan 03 2019, extended Jul 06 2019

A097940 Sum of smallest parts (counted with multiplicity) of all compositions of n.

Original entry on oeis.org

1, 4, 8, 20, 37, 86, 173, 372, 788, 1680, 3550, 7554, 15994, 33820, 71374, 150376, 316151, 663474, 1389760, 2906116, 6066899, 12645608, 26318870, 54700044, 113536171, 235363832, 487342781, 1007969620, 2082597193, 4298660754, 8864505305, 18263797648, 37597869188

Offset: 1

Vladeta Jovovic, Sep 05 2004

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Knopfmacher, Arnold; Munagi, Augustine O. Smallest parts in compositions, Kotsireas, Ilias S. (ed.) et al., Advances in combinatorics. 3rd Waterloo workshop on computer algebra (WWCA, W80) 2011, Waterloo, Canada, May 26-29, 2011. Berlin: Springer. 197-207 (2013).

Cf. A046746, A092309.

Drop[ CoefficientList[ Series[(1 - x)^2*Sum[k*x^k/(1 - x - x^k)^2, {k, 50}], {x, 0, 30}], x], 1] (* Robert G. Wilson v, Sep 08 2004 *)

G.f.: (1-x)^2 * Sum_{k>=1} k*x^k/(1-x-x^k)^2.
a(n) ~ n * 2^(n-3). - Vaclav Kotesovec, Sep 05 2014
a(n) = Sum_{k=1..n} A308630(n,k). - R. J. Mathar, Jun 12 2019

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

A196025 Total sum of parts greater than 1 in all the partitions of n except one copy of the smallest part greater than 1 of every partition.

Original entry on oeis.org

0, 0, 0, 2, 5, 16, 30, 63, 108, 189, 298, 483, 720, 1092, 1582, 2297, 3225, 4551, 6244, 8592, 11590, 15622, 20741, 27536, 36066, 47198, 61150, 79077, 101391, 129808, 164934, 209213, 263745, 331807, 415229, 518656, 644719, 799926, 988432, 1218979

Offset: 1

Omar E. Pol, Oct 27 2011

nonn

Also partial sums of A182709. Total sum of emergent parts in all partitions of all numbers <= n.

Also total sum of parts of all regions of n that do not contain 1 as a part (Cf. A083751, A187219). - Omar E. Pol, Mar 04 2012

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A026905, A046746, A066186, A135010, A138121, A182699, A182707, A182709, A183152, A193827, A196039, A196930, A196931, A198381.

a(n) = A066186(n) - A196039(n).

a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)). - Vaclav Kotesovec, Jul 06 2019

A196930 Triangle read by rows in which row n lists in nondecreasing order the smallest part of every partition of n that do not contain 1 as a part, with a(1) = 1.

Original entry on oeis.org

1, 2, 3, 2, 4, 2, 5, 2, 2, 3, 6, 2, 2, 3, 7, 2, 2, 2, 2, 3, 4, 8, 2, 2, 2, 2, 3, 3, 4, 9, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 5, 10, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 4, 5, 11, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 6, 12, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5, 6, 13

Offset: 1

Omar E. Pol, Oct 21 2011

For n >= 2, row n lists the parts of the head of the last section of the set of partitions of n, except the emergent parts.

Also 1 together with the integers > 1 of A196931.

			Written as a triangle:
1,
2,
3,
2,4,
2,5,
2,2,3,6
2,2,3,7,
2,2,2,2,3,4,8,
2,2,2,2,3,3,4,9,
2,2,2,2,2,2,2,3,3,4,5,10,
2,2,2,2,2,2,2,2,3,3,3,4,5,11,
2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,5,6,12,
2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,6,13,
...
Row n has length A002865(n), n >= 2. The sum of row n is A182708(n), n >= 2. The number of 2's in row n is A002865(n-2), n >= 4. Right border of triangle gives A000027.

Alois P. Heinz, Rows n = 1..33, flattened

Where records occur give A000041.

Cf. A002865, A046746, A135010, A138121, A141285, A182699, A183152, A193827, A196931.

p:= (f, g)-> zip((x, y)->x+y, f, g, 0):
b:= proc(n, i) option remember; local g, j, r;
      if n=0 then [1] elif i<2 then [0]
    else r:= b(n, i-1);
         for j to n/i do g:= b(n-i*j, i-1);
           r:= p(p(r, [0$i, g[1]]), subsop(1=0, g));
         od; r
      fi
    end:
T:= proc(n) local l; l:= b(n$2);
      `if`(n=1, 1, seq(i$l[i+1], i=2..nops(l)-1))
    end:
seq(T(n), n=1..16);  # Alois P. Heinz, May 30 2013

p[f_, g_] := Plus @@ PadRight[{f, g}]; b[n_, i_] := b[n, i] = Module[{ g, j, r}, Which[n == 0, {1}, i<2, {0}, True, r = b[n, i-1]; For[j = 1, j <= n/i, j++, g = b[n-i*j, i-1]; r = p[p[r, Append[Array[0&, i], g // First]], ReplacePart[g, 1 -> 0]]]; r]]; T[n_] := Module[{l}, l = b[n, n]; If[n == 1, {1}, Table[Array[i&, l[[i+1]]], {i, 2, Length[l]-1}] // Flatten]]; Table[T[n], {n, 1, 16}] // Flatten (* Jean-François Alcover, Jan 30 2014, after Alois P. Heinz *)

A206283 Triangle read by rows: T(n,k) = sum of the k-th parts of all partitions of n with their parts written in nondecreasing order.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Feb 13 2012

In row n, the sum of all odd-indexed terms minus the sum of all even-indexed terms is equal to A194714(n).

Reversed rows converge to A014153. - Alois P. Heinz, Feb 13 2012

			Row 4 is 9, 7, 3, 1 because the five partitions of 4, with their parts written in nondecreasing order, are
.                               4
.                               1, 3
.                               2, 2
.                               1, 1, 2
.                               1, 1, 1, 1
-------------------------------------------
And the sums of the columns are 9, 7, 3, 1.
.
Triangle begins:
   1;
   3,  1;
   5,  3,  1;
   9,  7,  3,  1;
  12, 12,  7,  3,  1;
  20, 21, 14,  7,  3,  1;
  25, 31, 24, 14,  7,  3,  1;
  38, 47, 40, 26, 14,  7,  3,  1;
  49, 66, 61, 43, 26, 14,  7,  3,  1;
  69, 93, 92, 70, 45, 26, 14,  7,  3,  1;

Alois P. Heinz, Rows n = 1..70, flattened

Column 1 is A046746. Row sums give A066186.

Cf. A014153, A181187, A194714.

More terms from Alois P. Heinz, Feb 13 2012

A210953 Triangle read by rows: T(n,k) = sum of all parts in the k-th column of the shell model of partitions considering only the n-th shell and with its parts aligned to the right margin.

Original entry on oeis.org

1, 0, 3, 0, 0, 5, 0, 0, 2, 9, 0, 0, 0, 3, 12, 0, 0, 0, 2, 9, 20, 0, 0, 0, 0, 3, 11, 25, 0, 0, 0, 0, 2, 9, 22, 38, 0, 0, 0, 0, 0, 3, 14, 28, 49, 0, 0, 0, 0, 0, 2, 9, 26, 44, 69, 0, 0, 0, 0, 0, 0, 3, 14, 37, 55, 87, 0, 0, 0, 0, 0, 0, 2, 9, 29, 62, 83, 123

Offset: 1

Omar E. Pol, Apr 22 2012

			For n = 6 and k = 1..6 the 6th shell looks like this:
-------------------------
k: 1,  2,  3,  4,  5,  6
-------------------------
.                      6
.                  3 + 3
.                  4 + 2
.              2 + 2 + 2
.                      1
.                      1
.                      1
.                      1
.                      1
.                      1
.                      1
.
The sums of all parts in columns 1-6 are
.  0,  0,  0,  2,  9, 20, the same as the 6th row of triangle.
Triangle begins:
1;
0, 3;
0, 0, 5;
0, 0, 2, 9;
0, 0, 0, 3, 12;
0, 0, 0, 2,  9, 20;
0, 0, 0, 0,  3, 11, 25;
0, 0, 0, 0,  2,  9, 22, 38;
0, 0, 0, 0,  0,  3, 14, 28, 49;
0, 0, 0, 0,  0,  2,  9, 26, 44, 69;
0, 0, 0, 0,  0,  0,  3, 14, 37, 55, 87;
0, 0, 0, 0,  0,  0,  2,  9, 29, 62, 83, 123;

Row sums give A138879. Column sums converge to A014153. Right border gives A046746, n >= 1.

Cf. A135010, A138121, A210950, A210951, A210952.

cp's OEIS Frontend

A336902 Sum of the smallest parts of all compositions of n into distinct parts.

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A336903 Sum of the largest parts of all compositions of n into distinct parts.

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A097147 Total sum of minimum block sizes in all partitions of n-set.

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A097148 Total sum of maximum block sizes in all partitions of n-set.

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A182707 Sum of the parts of all partitions of n-1 plus the sum of the emergent parts of the partitions of n.

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A097940 Sum of smallest parts (counted with multiplicity) of all compositions of n.

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A196025 Total sum of parts greater than 1 in all the partitions of n except one copy of the smallest part greater than 1 of every partition.

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A196930 Triangle read by rows in which row n lists in nondecreasing order the smallest part of every partition of n that do not contain 1 as a part, with a(1) = 1.

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