cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 21-26 of 26 results.

A360954 Number of finite sets of positive integers whose right half (exclusive) sums to n.

Original entry on oeis.org

1, 0, 1, 3, 6, 10, 15, 22, 29, 41, 50, 70, 81, 113, 126, 176, 191, 264, 286, 389, 413, 569, 595, 798, 861, 1121, 1187, 1585, 1653, 2132, 2334, 2906, 3111, 4006, 4234, 5252, 5818, 6995, 7620, 9453, 10102, 12165, 13663, 15940, 17498, 21127, 22961, 26881, 30222, 34678, 38569
Offset: 0

Views

Author

Gus Wiseman, Mar 09 2023

Keywords

Examples

			The a(2) = 1 through a(7) = 22 sets:
  {1,2}  {1,3}    {1,4}    {1,5}    {1,6}    {1,7}
         {2,3}    {2,4}    {2,5}    {2,6}    {2,7}
         {1,2,3}  {3,4}    {3,5}    {3,6}    {3,7}
                  {1,2,4}  {4,5}    {4,6}    {4,7}
                  {1,3,4}  {1,2,5}  {5,6}    {5,7}
                  {2,3,4}  {1,3,5}  {1,2,6}  {6,7}
                           {1,4,5}  {1,3,6}  {1,2,7}
                           {2,3,5}  {1,4,6}  {1,3,7}
                           {2,4,5}  {1,5,6}  {1,4,7}
                           {3,4,5}  {2,3,6}  {1,5,7}
                                    {2,4,6}  {1,6,7}
                                    {2,5,6}  {2,3,7}
                                    {3,4,6}  {2,4,7}
                                    {3,5,6}  {2,5,7}
                                    {4,5,6}  {2,6,7}
                                             {3,4,7}
                                             {3,5,7}
                                             {3,6,7}
                                             {4,5,7}
                                             {4,6,7}
                                             {5,6,7}
                                             {1,2,3,4}
For example, the set y = {1,2,3,4} has right half (exclusive) {3,4}, with sum 7, so y is counted under a(7).

Links

Crossrefs

The version for multisets is A360673, inclusive A360671.
The inclusive version is A360955.
First for prime indices, second for partitions, third for prime factors:
- A360676 gives left sum (exclusive), counted by A360672, product A361200.
- A360677 gives right sum (exclusive), counted by A360675, product A361201.
- A360678 gives left sum (inclusive), counted by A360675, product A347043.
- A360679 gives right sum (inclusive), counted by A360672, product A347044.
Cf. A000009, A072233, A359893, A359901, A360956.

Programs

  • Mathematica
    Table[Length[Select[Join@@IntegerPartitions/@Range[0,3*k],UnsameQ@@#&&Total[Take[#,Floor[Length[#]/2]]]==k&]],{k,0,15}]
  • PARI
    \\ P(n,k) is A072233(n,k).
P(n,k)=polcoef(1/prod(k=1, k, 1 - x^k + O(x*x^n)), n)
a(n)=if(n==0, 1, sum(w=1, sqrt(n), my(t=binomial(w,2)); sum(h=w+1, (n-t)\w, binomial(h, w+1) * P(n-w*h-t, w-1)))) \\ Andrew Howroyd, Mar 13 2023

Formula

a(n) = Sum_{w>=1} Sum_{h=w+1..floor((n-binomial(w,2))/w)} binomial(h,w+1) * A072233(n - w*h - binomial(w,2), w-1) for n > 0. - Andrew Howroyd, Mar 13 2023

Extensions

Terms a(16) and beyond from Andrew Howroyd, Mar 13 2023

A360955 Number of finite sets of positive integers whose right half (inclusive) sums to n.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 7, 11, 12, 19, 20, 31, 33, 49, 51, 77, 79, 112, 124, 165, 177, 247, 260, 340, 388, 480, 533, 693, 747, 925, 1078, 1271, 1429, 1772, 1966, 2331, 2705, 3123, 3573, 4245, 4737, 5504, 6424, 7254, 8256, 9634, 10889, 12372, 14251, 16031, 18379
Offset: 0

Views

Author

Gus Wiseman, Mar 09 2023

Keywords

Examples

			The a(1) = 1 through a(8) = 12 sets:
  {1}  {2}    {3}    {4}    {5}      {6}      {7}        {8}
       {1,2}  {1,3}  {1,4}  {1,5}    {1,6}    {1,7}      {1,8}
              {2,3}  {2,4}  {2,5}    {2,6}    {2,7}      {2,8}
                     {3,4}  {3,5}    {3,6}    {3,7}      {3,8}
                            {4,5}    {4,6}    {4,7}      {4,8}
                            {1,2,3}  {5,6}    {5,7}      {5,8}
                                     {1,2,4}  {6,7}      {6,8}
                                              {1,2,5}    {7,8}
                                              {1,3,4}    {1,2,6}
                                              {2,3,4}    {1,3,5}
                                              {1,2,3,4}  {2,3,5}
                                                         {1,2,3,5}
For example, the set y = {2,3,5} has right half (inclusive) {3,5}, with sum 8, so y is counted under a(8).

Links

Crossrefs

The version for multisets is A360671, exclusive A360673.
The exclusive version is A360954.
First for prime indices, second for partitions, third for prime factors:
- A360676 gives left sum (exclusive), counted by A360672, product A361200.
- A360677 gives right sum (exclusive), counted by A360675, product A361201.
- A360678 gives left sum (inclusive), counted by A360675, product A347043.
- A360679 gives right sum (inclusive), counted by A360672, product A347044.
Cf. A000009, A072233, A359893, A359901, A360674, A360956.

Programs

  • Mathematica
    Table[Length[Select[Join@@IntegerPartitions/@Range[0,3*k], UnsameQ@@#&&Total[Take[#,Ceiling[Length[#]/2]]]==k&]],{k,0,15}]
  • PARI
    \\ P(n,k) is A072233(n,k).
P(n,k)=polcoef(1/prod(k=1, k, 1 - x^k + O(x*x^n)), n)
a(n)=if(n==0, 1, sum(w=1, sqrt(n), my(t=binomial(w,2)); sum(h=w, (n-t)\w, binomial(h, w) * P(n-w*h-t, w-1)))) \\ Andrew Howroyd, Mar 13 2023

Formula

a(n) = Sum_{w>=1} Sum_{h=w..floor((n-binomial(w,2))/w)} binomial(h,w) * A072233(n - w*h - binomial(w,2), w-1) for n > 0. - Andrew Howroyd, Mar 13 2023

Extensions

Terms a(16) and beyond from Andrew Howroyd, Mar 13 2023

A360956 Number of finite even-length multisets of positive integers whose right half sums to n.

Original entry on oeis.org

1, 1, 3, 5, 10, 13, 26, 31, 55, 73, 112, 140, 233, 276, 405, 539, 750, 931, 1327, 1627, 2259, 2839, 3708, 4624, 6237, 7636, 9823, 12275, 15715, 19227, 24735, 30000, 37930, 46339, 57574, 70374, 87704, 105606, 129998, 157417, 193240, 231769, 283585, 339052, 411682, 493260
Offset: 0

Views

Author

Gus Wiseman, Mar 09 2023

Keywords

Examples

			The a(1) = 1 through a(5) = 13 multisets:
  {1,1}  {1,2}      {1,3}          {1,4}              {1,5}
         {2,2}      {2,3}          {2,4}              {2,5}
         {1,1,1,1}  {3,3}          {3,4}              {3,5}
                    {1,1,1,2}      {4,4}              {4,5}
                    {1,1,1,1,1,1}  {1,1,1,3}          {5,5}
                                   {1,1,2,2}          {1,1,1,4}
                                   {1,2,2,2}          {1,1,2,3}
                                   {2,2,2,2}          {1,2,2,3}
                                   {1,1,1,1,1,2}      {2,2,2,3}
                                   {1,1,1,1,1,1,1,1}  {1,1,1,1,1,3}
                                                      {1,1,1,1,2,2}
                                                      {1,1,1,1,1,1,1,2}
                                                      {1,1,1,1,1,1,1,1,1,1}
For example, the multiset y = {1,2,2,3} has right half {2,3}, with sum 5, so y is counted under a(5).

Links

Crossrefs

This is the even-length case of A360671 and A360673.
First for prime indices, second for partitions, third for prime factors:
- A360676 gives left sum (exclusive), counted by A360672, product A361200.
- A360677 gives right sum (exclusive), counted by A360675, product A361201.
- A360678 gives left sum (inclusive), counted by A360675, product A347043.
- A360679 gives right sum (inclusive), counted by A360672, product A347044.
Cf. A000041, A360674, A360954, A360955.

Programs

  • Mathematica
    Table[Length[Select[Join@@IntegerPartitions/@Range[0,3*k], EvenQ[Length[#]]&&Total[Take[#,Length[#]/2]]==k&]],{k,0,15}]
  • PARI
    seq(n)={my(s=1 + O(x*x^n), p=s); for(k=1, n, s += p*x^k/(1-x^k + O(x*x^(n-k)))^(k+1); p /= 1 - x^k); Vec(s)} \\ Andrew Howroyd, Mar 11 2023

Formula

G.f.: 1 + Sum_{k>=1} x^k/((1 - x^k)^(k+1) * Product_{j=1..k-1} (1-x^j)). - Andrew Howroyd, Mar 11 2023

Extensions

Terms a(16) and beyond from Andrew Howroyd, Mar 11 2023

A362558 Number of integer partitions of n without a nonempty initial consecutive subsequence summing to n/2.

Original entry on oeis.org

1, 1, 1, 3, 2, 7, 6, 15, 11, 30, 27, 56, 44, 101, 93, 176, 149, 297, 271, 490, 432, 792, 744, 1255, 1109, 1958, 1849, 3010, 2764, 4565, 4287, 6842, 6328, 10143, 9673, 14883, 13853, 21637, 20717, 31185, 29343, 44583, 42609, 63261, 60100, 89134, 85893, 124754
Offset: 0

Views

Author

Gus Wiseman, Apr 24 2023

Keywords

Comments

Also the number of n-multisets of positive integers that (1) have integer median, (2) cover an initial interval, and (3) have weakly decreasing multiplicities.

Examples

			The a(1) = 1 through a(7) = 15 partitions:
  (1)  (2)  (3)    (4)   (5)      (6)     (7)
            (21)   (31)  (32)     (42)    (43)
            (111)        (41)     (51)    (52)
                         (221)    (222)   (61)
                         (311)    (411)   (322)
                         (2111)   (2211)  (331)
                         (11111)          (421)
                                          (511)
                                          (2221)
                                          (3211)
                                          (4111)
                                          (22111)
                                          (31111)
                                          (211111)
                                          (1111111)
The partition y = (3,2,1,1,1) has nonempty initial consecutive subsequences (3,2,1,1,1), (3,2,1,1), (3,2,1), (3,2), (3), with sums 8, 7, 6, 5, 3. Since 4 is missing, y is counted under a(8).

Crossrefs

The odd bisection is A058695.
The version for compositions is A213173.
The complement is counted by A322439 aerated.
The even bisection is A362051.
For mean instead of median we have A362559.
A000041 counts integer partitions, strict A000009.
A325347 counts partitions with integer median, complement A307683.
A359893/A359901/A359902 count partitions by median.
Cf. A058398, A108917, A169942, A325676, A353864, A360254, A360672, A360675, A360686, A360687, A362560.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],!MemberQ[Accumulate[#],n/2]&]],{n,0,15}]

A363527 Number of integer partitions of n with weighted sum 3*n.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 3, 4, 4, 6, 8, 7, 10, 13, 13, 21, 25, 24, 37, 39, 40, 58, 63, 72, 94, 106, 118, 144, 165, 181, 224, 256, 277, 341, 387, 417, 504, 560, 615, 743, 818, 899, 1066, 1171, 1285, 1502, 1655, 1819, 2108, 2315, 2547, 2915
Offset: 0

Views

Author

Gus Wiseman, Jun 11 2023

Keywords

Comments

Are the partitions counted all of length > 4?
The (one-based) weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i. The reverse-weighted sum is the weighted sum of the reverse, also the sum of partial sums. For example, the weighted sum of (4,2,2,1) is 1*4 + 2*2 + 3*2 + 4*1 = 18 and the reverse-weighted sum is 4*4 + 3*2 + 2*2 + 1*1 = 27.

Examples

			The partition (2,2,1,1,1,1) has sum 8 and weighted sum 24 so is counted under a(8).
The a(13) = 1 through a(18) = 8 partitions:
  (332221)  (333221)    (33333)     (442222)    (443222)    (443331)
            (4322111)   (522222)    (5322211)   (4433111)   (444222)
            (71111111)  (4332111)   (55111111)  (5332211)   (533322)
                        (63111111)  (63211111)  (55211111)  (4443111)
                                                (63311111)  (7222221)
                                                (72221111)  (55311111)
                                                            (64221111)
                                                            (A11111111)

Crossrefs

The version for compositions is A231429.
The reverse version is A363526.
These partitions have ranks A363531.
A000041 counts integer partitions, strict A000009.
A053632 counts compositions by weighted sum, rank statistic A029931/A359042.
A264034 counts partitions by weighted sum, reverse A358194.
A304818 gives weighted sum of prime indices, row-sums of A359361.
A318283 gives weighted sum of reversed prime indices, row-sums of A358136.
A320387 counts multisets by weighted sum, zero-based A359678.
Cf. A000016, A008284, A067538, A222855, A222970, A359755, A360672, A360675, A362559, A362560, A363525, A363528, A363532.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Total[Accumulate[Reverse[#]]]==3n&]],{n,0,30}]

A362051 Number of integer partitions of 2n without a nonempty initial consecutive subsequence summing to n.

Original entry on oeis.org

1, 1, 2, 6, 11, 27, 44, 93, 149, 271, 432, 744, 1109, 1849, 2764, 4287, 6328, 9673, 13853, 20717, 29343, 42609, 60100, 85893, 118475, 167453, 230080, 318654, 433763, 595921, 800878, 1090189, 1456095, 1957032, 2600199, 3465459, 4558785, 6041381, 7908681
Offset: 0

Views

Author

Gus Wiseman, Apr 24 2023

Keywords

Comments

Even bisection of A362558.
a(0) = 1; a(n) = A000041(2n) - A322439(n). - Alois P. Heinz, Apr 27 2023

Examples

			The a(1) = 1 through a(4) = 11 partitions:
  (2)  (4)   (6)     (8)
       (31)  (42)    (53)
             (51)    (62)
             (222)   (71)
             (411)   (332)
             (2211)  (521)
                     (611)
                     (3221)
                     (3311)
                     (5111)
                     (32111)
The partition y = (3,2,1,1,1) has nonempty initial consecutive subsequences (3,2,1,1,1), (3,2,1,1), (3,2,1), (3,2), (3), with sums 8, 7, 6, 5, 3. Since 4 is missing, y is counted under a(4).

Crossrefs

The version for compositions is A000302, bisection of A213173.
The complement is counted by A322439.
Even bisection of A362558.
A000041 counts integer partitions, strict A000009.
A304442 counts partitions with all equal run-sums.
A325347 counts partitions with integer median, complement A307683.
A353836 counts partitions by number of distinct run-sums.
A359893/A359901/A359902 count partitions by median.
Cf. A108917, A169942, A237363, A325676, A353864, A360254, A360672, A360675, A360686, A360952, A362560.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[2n],!MemberQ[Accumulate[#],n]&]],{n,0,15}]
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