A194552 -id:A194552 - cp's OEIS frontend

A138880 Sum of all parts of all partitions of n that do not contain 1 as a part.

0, 2, 3, 8, 10, 24, 28, 56, 72, 120, 154, 252, 312, 476, 615, 880, 1122, 1584, 1995, 2740, 3465, 4620, 5819, 7680, 9575, 12428, 15498, 19824, 24563, 31170, 38378, 48224, 59202, 73678, 90055, 111384, 135420, 166364, 201630, 246120, 297045, 360822

Offset: 1

Omar E. Pol, Apr 30 2008

nonn

Sum of all parts > 1 of the last section of the set of partitions of n.
Row sums of triangle A182710. Also row sums of other similar triangles as A138136 and A182711.
Partial sums give A194552. - Omar E. Pol, Sep 23 2013

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

Cf. A000041, A002865, A066186, A133041, A138135, A138136, A138137, A138138, A138151, A138879, A139100.

Table[Total[Flatten[Select[IntegerPartitions[n],FreeQ[#,1]&]]],{n,50}] (* Harvey P. Dale, May 24 2015 *)
a[n_] := (PartitionsP[n] - PartitionsP[n-1])*n; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Oct 07 2015 *)

a(n) = A002865(n)*n = (A000041(n) - A000041(n-1))*n = A138879(n) - A000041(n-1).
a(n) ~ Pi^2/6*A000070(n-2). - Peter Bala, Dec 23 2013
G.f.: x*f'(x), where f(x) = Product_{k>=2} 1/(1 - x^k). - Ilya Gutkovskiy, Apr 13 2017
a(n) ~ Pi * exp(sqrt(2*n/3)*Pi) / (12*sqrt(2*n)) * (1 - (3*sqrt(3/2)/Pi + 13*Pi/(24*sqrt(6)))/sqrt(n) + (217*Pi^2/6912 + 9/(2*Pi^2) + 13/8)/n). - Vaclav Kotesovec, Jul 06 2019

Better definition from Omar E. Pol, Sep 23 2013

A206561 Triangle read by rows: T(n,k) = total sum of parts >= k in all partitions of n.

Original entry on oeis.org

1, 4, 2, 9, 5, 3, 20, 13, 7, 4, 35, 23, 15, 9, 5, 66, 47, 31, 19, 11, 6, 105, 75, 53, 35, 23, 13, 7, 176, 131, 93, 66, 42, 27, 15, 8, 270, 203, 151, 106, 74, 49, 31, 17, 9, 420, 323, 241, 178, 126, 86, 56, 35, 19, 10, 616, 477, 365, 272, 200, 140, 98, 63, 39, 21, 11

Offset: 1

Omar E. Pol, Feb 14 2012

From Omar E. Pol, Mar 18 2018: (Start)
In the n-th row of the triangle the first differences together with its last term give the n-th row of triangle A138785 (see below):
Row..........:  1     2       3            4               5          ...
               ---  ----   -------   ------------   ----------------
This triangle:  1;  4, 2;  9, 5, 3;  20, 13, 7, 4;  35, 23, 15, 9, 5; ...
                |   | /|   | /| /|    | / | /| /|    | / | / | /| /|
                |   |/ |   |/ |/ |    |/  |/ |/ |    |/  |/  |/ |/ |
A138785......:  1;  2, 2;  4, 2, 3;   7,  6, 3, 4;  12,  8,  6, 4, 5; ... (End)

			Triangle begins:
    1;
    4,  2;
    9,  5,  3;
   20, 13,  7,  4;
   35, 23, 15,  9,  5;
   66, 47, 31, 19, 11,  6;
  105, 75, 53, 35, 23, 13,  7;
  ...

Alois P. Heinz, Rows n = 1..141, flattened

Columns 1-2 give A066186, A194552.
Right border gives A000027.
Cf. A138785, A181187.
Row sums give A066183. - Omar E. Pol, Mar 19 2018
Both A180681 and A299768 have the same row sums as this triangle. - Omar E. Pol, Mar 21 2018

Table[With[{s = IntegerPartitions[n]}, Table[Total@ Flatten@ Map[Select[#, # >= k &] &, s], {k, n}]], {n, 11}] // Flatten (* Michael De Vlieger, Mar 19 2018 *)

T(n,n) = n, T(n,k) = T(n,k+1) + k * A066633(n,k) for k < n.

T(n,k) = Sum_{i=k..n} A138785(n,i).

More terms from Alois P. Heinz, Feb 14 2012

A228816 Sum of all parts of all partitions of n that contain 1 as a part.

Original entry on oeis.org

1, 2, 6, 12, 25, 42, 77, 120, 198, 300, 462, 672, 1001, 1414, 2025, 2816, 3927, 5346, 7315, 9800, 13167, 17424, 23046, 30120, 39375, 50908, 65772, 84280, 107822, 136950, 173724, 218944, 275517, 344862, 430850, 535788, 665149, 822206, 1014585, 1247400

Offset: 1

Omar E. Pol, Sep 23 2013

nonn

Column 1 of A182701.

Cf. A000041, A000070, A066186, A138880, A194552, A228823.

Table[Total[Flatten[Select[IntegerPartitions[n],MemberQ[#,1]&]]],{n,40}] (* Harvey P. Dale, Sep 27 2015 *)

a(n) = n*A000041(n-1).
a(n) = A066186(n-1) + A000041(n-1), n >= 2.
a(n) = A194552(n-1) + A000070(n-1), n >= 2.

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A138880 Sum of all parts of all partitions of n that do not contain 1 as a part.

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A206561 Triangle read by rows: T(n,k) = total sum of parts >= k in all partitions of n.

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A228816 Sum of all parts of all partitions of n that contain 1 as a part.

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