A046746 -id:A046746 - cp's OEIS frontend

A286014 Sum of smallest parts of all partitions of n into consecutive parts.

1, 2, 4, 4, 7, 7, 10, 8, 15, 11, 16, 15, 19, 16, 27, 16, 25, 26, 28, 22, 38, 26, 34, 31, 40, 31, 50, 29, 43, 49, 46, 32, 62, 41, 59, 48, 55, 46, 74, 46, 61, 67, 64, 46, 94, 56, 70, 63, 77, 69, 98, 55, 79, 85, 92, 61, 110, 71, 88, 93, 91, 76, 131, 64, 110, 103

Offset: 1

Omar E. Pol, Apr 30 2017

nonn

If n is a power of 2 then a(n) = n, the same as A286015(n).

Conjecture: this is also the row sums of A211343.

			For n = 15 there are four partitions of 15 into consecutive parts: [15], [8, 7], [6, 5, 4] and [5, 4, 3, 2, 1]. The sum of the smallest parts is 15 + 7 + 4 + 1 = 27, so a(15) = 27.

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

Cf. A000079, A046746, A204217, A211343, A245579, A286015.

Table[Total[Select[IntegerPartitions@ n, Or[Length@ # == 1, Union@ Differences@ # == {-1}] &][[All, -1]]], {n, 66}] (* Michael De Vlieger, Jul 21 2017 *)

More terms from Alois P. Heinz, May 01 2017

A286015 Sum of largest parts of all partitions of n into consecutive parts.

Original entry on oeis.org

1, 2, 5, 4, 8, 9, 11, 8, 18, 14, 17, 17, 20, 19, 34, 16, 26, 31, 29, 26, 46, 29, 35, 33, 45, 34, 58, 35, 44, 58, 47, 32, 70, 44, 70, 57, 56, 49, 82, 50, 62, 78, 65, 53, 114, 59, 71, 65, 84, 76, 106, 62, 80, 98, 106, 67, 118, 74, 89, 106, 92, 79, 153, 64, 124

Offset: 1

Omar E. Pol, Apr 30 2017

nonn

If n is a power of 2 then a(n) = n, the same as A286014(n).

Conjecture: this is also the row sums of A286013.

			For n = 15 there are four partitions of 15 into consecutive parts: [15], [8, 7], [6, 5, 4] and [5, 4, 3, 2, 1]. The sum of the largest parts is 15 + 8 + 6 + 5 = 34, so a(15) = 34.

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

Cf. A000079, A006128, A046746, A204217, A211343, A245579, A286013, A286014.

Table[Total[Select[IntegerPartitions@ n, Or[Length@ # == 1, Union@ Differences@ # == {-1}] &][[All, 1]]], {n, 65}] (* Michael De Vlieger, Jul 21 2017 *)

More terms from Alois P. Heinz, May 01 2017

A097145 Total sum of minimum list sizes in all sets of lists of n-set, cf. A000262.

Original entry on oeis.org

0, 1, 5, 25, 157, 1101, 9211, 85513, 900033, 10402633, 133059331, 1836961941, 27619253113, 444584808253, 7678546353843, 140944884572521, 2751833492404321, 56691826303303953, 1233793951629951043, 28191548364561422173, 676190806704598883241

Offset: 0

Vladeta Jovovic, Jul 27 2004

			For n=4 we have 73 sets of lists (cf. A000262): (1234) (24 ways), (123)(4) (6*4 ways), (12)(34) (3*4 ways), (12)(3)(4) (6*2 ways), (1)(2)(3)(4) (1 way); so a(n)= 24*4+24*1+12*2+12*1+1*1 = 157.

Alois P. Heinz, Table of n, a(n) for n = 0..444

Cf. A000262, A028417, A028418, A046746, A006128, A097146-A097148.

b:= proc(n, m) option remember; `if`(n=0, m, add(j!*
      b(n-j, min(m, j))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> `if`(n=0, 0, b(n, infinity)):
seq(a(n), n=0..25);  # Alois P. Heinz, May 10 2016

b[n_, m_] := b[n, m] = If[n==0, m, Sum[j!*b[n-j, Min[m, j]]*Binomial[n-1, j - 1], {j, 1, n}]]; a[n_] := If[n==0, 0, b[n, Infinity]]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 18 2017, after Alois P. Heinz *)

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

More terms from Max Alekseyev, Jul 04 2009

a(0)=0 prepended by Alois P. Heinz, May 10 2016

A182708 a(n) is the sum of the smallest parts of all partitions of n that do not contain 1 as a part.

Original entry on oeis.org

0, 2, 3, 6, 7, 13, 14, 23, 27, 39, 45, 67, 75, 104, 125, 165, 194, 258, 302, 392, 467, 588, 700, 885, 1045, 1296, 1546, 1897, 2249, 2753, 3252, 3945, 4670, 5616, 6633, 7957, 9357, 11157, 13124, 15573, 18257, 21599, 25259, 29760, 34760, 40788, 47526, 55642, 64669, 75465, 87576, 101898, 117991, 136977, 158286

Offset: 1

Omar E. Pol, Nov 28 2010

In other words, sum of the smallest parts of all partitions of the head of the last section of the set of partitions of n.

Only one of the smallest parts is used in the sum.

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

Cf. A000041, A046746, A135010, A138121, A182707, A182709.

Table[Total[{Min /@ IntegerPartitions[n, All, Range[2, n]]}, 2], {n, 55}] (* Robert Price, Aug 30 2020 *) (* Only suitable for n<100 *)

my(N=66, z='z+O('z^N));  gf=sum(k=1, N, k * z^k / prod(j=k, N, 1-z^j ) ) - z/eta(z); concat([0], Vec(gf)) \\ Joerg Arndt, Aug 31 2020

a(n) = A046746(n) - A000041(n-1).

a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (6*sqrt(2)*n^(3/2)) * (1 + (11*Pi/(24*sqrt(6)) - 3*sqrt(3/2)/Pi)/sqrt(n)). - Vaclav Kotesovec, Jan 03 2019, extended Jul 06 2019

A182977 Total number of parts that are neither the smallest part nor the largest part in all partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 2, 6, 12, 22, 39, 66, 103, 159, 243, 352, 510, 721, 1011, 1391, 1903, 2557, 3436, 4549, 5999, 7824, 10187, 13132, 16886, 21544, 27414, 34657, 43703, 54797, 68558, 85328, 105963, 131028, 161664, 198710

Offset: 0

Omar E. Pol, Jul 17 2011

nonn

			For n = 6 the partitions of 6 are
6
5 + 1
4 + 2
4 + 1 + 1
3 + 3
3 + (2) + 1 .......... the "2" is the part that counts.
3 + 1 + 1 + 1
2 + 2 + 2
2 + 2 + 1 + 1
2 + 1 + 1 + 1 + 1
1 + 1 + 1 + 1 + 1 + 1
There is only one part which is neither the smallest part nor the largest part in all partitions of 6, so a(6) = 1.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A006128, A046746, A092269, A116686, A182978, A182984, A265249.

g := add(add((add(x^(i+j+k)/(1-x^k), k = i+1 .. j-1))/(mul(1-x^k, k = i .. j)), j = i+1 .. 80), i = 1 .. 80): gser := series(g, x = 0, 50): seq(coeff(gser, x, n), n = 0 .. 45); # Emeric Deutsch, Dec 25 2015

a(n) = A006128(n) - A182978(n).
G.f.: g(x) = Sum_{i>=1} Sum_{j>=i+1} (Sum_{k=i+1..j-1} x^{i+j+k}/(1-x^k)/Product_{k=i..j}(1-x^k)). - Emeric Deutsch, Dec 25 2015
a(n) = Sum_{k>=0} k*A265249(n,k). - Emeric Deutsch, Dec 25 2015

a(12) corrected and more terms a(13)-a(40) from David Scambler, Jul 18 2011

A097146 Total sum of maximum list sizes in all sets of lists of n-set, cf. A000262.

Original entry on oeis.org

0, 1, 5, 31, 217, 1781, 16501, 172915, 1998641, 25468777, 352751941, 5292123431, 85297925065, 1472161501981, 27039872306357, 527253067633531, 10865963240550241, 236088078855319505, 5390956470528548101, 129102989125943058607, 3234053809095307670201, 84596120521251178630981, 2305894874979300173268085

Offset: 0

Vladeta Jovovic, Jul 27 2004

			For n=4 we have 73 sets of lists (cf. A000262): (1234) (24 ways), (123)(4) (6*4 ways), (12)(34) (3*4 ways), (12)(3)(4) (6*2 ways), (1)(2)(3)(4) (1 way); so a(4)= 24*4+24*3+12*2+12*2+1*1 = 217.

Alois P. Heinz, Table of n, a(n) for n = 0..444

Cf. A028417, A028418, A046746, A006128, A097145, A097147, A097148.

b:= proc(n, m) option remember; `if`(n=0, m, add(j!*
      b(n-j, max(m, j))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..25);  # Alois P. Heinz, May 10 2016

b[n_, m_] := b[n, m] = If[n == 0, m, Sum[j! b[n-j, Max[m, j]] Binomial[n-1, j-1], {j, 1, n}]];
a[n_] := b[n, 0];
a /@ Range[0, 25] (* Jean-François Alcover, Nov 05 2020, after Alois P. Heinz *)

N=50; x='x+O('x^N);
egf=exp(x/(1-x))*sum(k=1,N, (1-exp(x^k/(x-1))) );
Vec( serlaplace(egf) ) /* show terms */

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

a(0)=0 prepended by Alois P. Heinz, May 10 2016

A182978 Total number of parts that are the smallest part or the largest part in all partitions of n.

Original entry on oeis.org

1, 3, 6, 12, 20, 34, 52, 80, 116, 170, 236, 333, 453, 621, 825, 1111, 1455, 1923, 2487, 3239, 4149, 5342, 6770, 8625, 10852, 13698, 17107, 21413, 26567, 33019, 40721, 50270, 61663, 75665, 92318, 112686, 136849, 166173, 200923, 242836

Offset: 1

Omar E. Pol, Jul 17 2011

nonn

			For n = 6 the partitions of 6 are
6
5 + 1
4 + 2
4 + 1 + 1
3 + 3
3 + (2) + 1 .... the "2" is the part that does not count.
3 + 1 + 1 + 1
2 + 2 + 2
2 + 2 + 1 + 1
2 + 1 + 1 + 1 + 1
1 + 1 + 1 + 1 + 1 + 1
The total number of parts in all partitions of 6 is equal to 35. All parts are the smallest part or the largest part, except the "2" in the partition (3 + 2 + 1), so a(6) = 35 - 1 = 34.

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

Cf. A006128, A046746, A092269, A116686, A182977, A182984.

l:= proc(n, i) option remember; `if`(n=i, n, 0)+
      `if`(i<1, 0, l(n, i-1) +`if`(n l(n, n) +s(n, n) -numtheory[sigma](n):
seq(a(n), n=1..50);  # Alois P. Heinz, Jan 17 2013

l[n_, i_] := l[n, i] = If[n==i, n, 0] + If[i<1, 0, l[n, i-1] + If[nJean-François Alcover, Nov 03 2015, after Alois P. Heinz *)

a(n) = A006128(n) - A182977(n).

a(12) corrected and more terms a(13)-a(40) from David Scambler, Jul 18 2011

A182629 Total number of largest parts in all partitions of n that contain at least two distinct parts.

Original entry on oeis.org

0, 0, 0, 1, 2, 6, 8, 17, 23, 36, 51, 75, 95, 138, 181, 236, 310, 407, 516, 667, 840, 1062, 1344, 1678, 2080, 2589, 3212, 3942, 4851, 5937, 7246, 8824, 10724, 12971, 15705, 18895, 22749, 27296, 32734, 39083, 46668, 55553, 66086, 78389, 92937, 109857, 129850

Offset: 0

Omar E. Pol, Jul 14 2011

nonn

a(n) is also the sum of smallest parts of all partitions of n minus the sum of divisors of n, for n >= 1.

			For n = 6 the partitions of 6 are
6 ....................... all parts are equal.
5 + 1 ................... contains only one largest part.
4 + 2 ................... contains only one largest part.
4 + 1 + 1 ............... contains only one largest part.
3 + 3 ................... all parts are equal.
3 + 2 + 1 ............... contains only one largest part.
3 + 1 + 1 + 1 ........... contains only one largest part.
2 + 2 + 2 ............... all parts are equal.
2 + 2 + 1 + 1 ........... contains two largest parts.
2 + 1 + 1 + 1 + 1 ....... contains only one largest part.
1 + 1 + 1 + 1 + 1 + 1 ... all parts are equal.
There are 8 largest parts, so a(6) = 8.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000203, A006128, A046746, A144300.

b:= proc(n, i) option remember; `if`(n=i, n, 0)+
      `if`(i<1, 0, b(n, i-1) +`if`(n b(n, n) -numtheory[sigma](n):
seq(a(n), n=0..100);  # Alois P. Heinz, Jan 17 2013

b[n_, i_] := b[n, i] = If[n == i, n, 0] + If[i < 1, 0, b[n, i - 1] + If[n < i, 0, b[n - i, i]]]; a[n_] := b[n, n] - DivisorSigma[1, n]; a[0] = 0; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 06 2017, after Alois P. Heinz *)

a(n) = A046746(n) - A000203(n), for n >= 1. - Omar E. Pol, Jul 15 2011

More terms a(13)-a(46) from David Scambler, Jul 15 2011

A196039 Total sum of the smallest part of every partition of every shell of n.

Original entry on oeis.org

0, 1, 4, 9, 18, 30, 50, 75, 113, 162, 231, 318, 441, 593, 798, 1058, 1399, 1824, 2379, 3066, 3948, 5042, 6422, 8124, 10264, 12884, 16138, 20120, 25027, 30994, 38312, 47168, 57955, 70974, 86733, 105676, 128516, 155850, 188644, 227783, 274541

Offset: 0

Omar E. Pol, Oct 27 2011

nonn

Partial sums of A046746.

Total sum of parts of all regions of n that contain 1 as a part. - Omar E. Pol, Mar 11 2012

			For n = 5 the seven partitions of 5 are:
5
3         + 2
4             + 1
2     + 2     + 1
3         + 1 + 1
2     + 1 + 1 + 1
1 + 1 + 1 + 1 + 1
.
The five shells of 5 (see A135010 and also A138121), written as a triangle, are:
1
2, 1
3, 1, 1
4, (2, 2), 1, 1, 1
5, (3, 2), 1, 1, 1, 1, 1
.
The first "2" of row 4 does not count, also the "3" of row 5 does not count, so we have:
1
2, 1
3, 1, 1
4, 2, 1, 1, 1
5, 2, 1, 1, 1, 1, 1
.
thus a(5) = 1+2+1+3+1+1+4+2+1+1+1+5+2+1+1+1+1+1 = 30.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
Omar E. Pol, Illustration of the seven regions of 5

Cf. A026905, A046746, A066186, A135010, A138121, A182699, A182707, A182709, A183152, A193827, A196025, A196930, A196931, A198381, A206437.

b:= proc(n, i) option remember;
     `if`(n=i, n, 0) +`if`(i<1, 0, b(n, i-1) +`if`(nAlois P. Heinz, Apr 03 2012

b[n_, i_] := b[n, i] = If[n == i, n, 0] + If[i < 1, 0, b[n, i-1] + If[n < i, 0, b[n-i, i]]]; Accumulate[Table[b[n, n], {n, 0, 50}]] (* Jean-François Alcover, Feb 05 2017, after Alois P. Heinz *)

a(n) = A066186(n) - A196025(n).

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

A225598 Triangle read by rows: T(n,k) = sum of all parts of all regions of the set of partitions of n whose largest part is k.

Original entry on oeis.org

1, 1, 3, 1, 3, 5, 1, 5, 5, 9, 1, 5, 8, 9, 12, 1, 7, 11, 15, 12, 20, 1, 7, 14, 19, 19, 20, 25, 1, 9, 17, 29, 24, 33, 25, 38, 1, 9, 23, 33, 36, 42, 39, 38, 49, 1, 11, 26, 47, 46, 61, 49, 61, 49, 69, 1, 11, 32, 55, 63, 76, 70, 76, 76, 69, 87, 1, 13, 38, 73, 78, 110, 87, 111, 95, 108, 87, 123

Offset: 1

Omar E. Pol, Aug 02 2013

For the definition of region see A206437.

T(n,k) is also the sum of all parts that end in the k-th column of the diagram of regions of the set of partitions of n (see Example section).

			For n = 5 and k = 3 the set of partitions of 5 contains two regions whose largest part is 3, they are third region which contains three parts [3, 1, 1] and the sixth region which contains only one part [3]. Therefore the sum of all parts is 3 + 1 + 1 + 3 = 8, so T(5,3) = 8.
.
.    Diagram    Illustration of parts ending in column k:
.    for n=5      k=1   k=2     k=3       k=4        k=5
.   _ _ _ _ _                                  _ _ _ _ _
.  |_ _ _    |                _ _ _           |_ _ _ _ _|
.  |_ _ _|_  |               |_ _ _|  _ _ _ _       |_ _|
.  |_ _    | |          _ _          |_ _ _ _|        |_|
.  |_ _|_  | |         |_ _|  _ _ _      |_ _|        |_|
.  |_ _  | | |          _ _  |_ _ _|       |_|        |_|
.  |_  | | | |      _  |_ _|     |_|       |_|        |_|
.  |_|_|_|_|_|     |_|   |_|     |_|       |_|        |_|
.
k = 1 2 3 4 5
.
The 5th row lists:  1     5       8         9         12
.
Triangle begins:
1;
1,  3;
1,  3,  5;
1,  5,  5,  9;
1,  5,  8,  9, 12;
1,  7, 11, 15, 12,  20;
1,  7, 14, 19, 19,  20, 25;
1,  9, 17, 29, 24,  33, 25,  38;
1,  9, 23, 33, 36,  42, 39,  38, 49;
1, 11, 26, 47, 46,  61, 49,  61, 49,  69;
1, 11, 32, 55, 63,  76, 70,  76, 76,  69, 87;
1, 13, 38, 73, 78, 110, 87, 111, 95, 108, 87, 123;

Column 1 is A000012. Column 2 are the numbers >= 3 of A109613. Row sums give A066186. Right border gives A046746. Second right border gives A046746.

Cf. A000041, A066186, A135010, A141285, A186114, A186412, A187219, A194446, A206437, A207779, A211978, A225597, A225600, A225610.

cp's OEIS Frontend

A286014 Sum of smallest parts of all partitions of n into consecutive parts.

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A286015 Sum of largest parts of all partitions of n into consecutive parts.

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A097145 Total sum of minimum list sizes in all sets of lists of n-set, cf. A000262.

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A182708 a(n) is the sum of the smallest parts of all partitions of n that do not contain 1 as a part.

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A182977 Total number of parts that are neither the smallest part nor the largest part in all partitions of n.

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A097146 Total sum of maximum list sizes in all sets of lists of n-set, cf. A000262.

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A182978 Total number of parts that are the smallest part or the largest part in all partitions of n.

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