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.

Showing 1-4 of 4 results.

A360671 Number of multisets whose right half (inclusive) sums to n.

Original entry on oeis.org

1, 2, 5, 8, 16, 21, 42, 51, 90, 121, 185, 235, 386, 465, 679, 908, 1261, 1580, 2238, 2770, 3827, 4831, 6314, 7910, 10619, 13074, 16813, 21049, 26934, 33072, 42445, 51679, 65264, 79902, 99309, 121548, 151325, 182697, 224873, 272625, 334536, 401999, 491560, 588723
Offset: 0

Views

Author

Gus Wiseman, Mar 09 2023

Keywords

Examples

			The a(0) = 1 through a(4) = 16 multisets:
  {}  {1}    {2}        {3}            {4}
      {1,1}  {1,2}      {1,3}          {1,4}
             {2,2}      {2,3}          {2,4}
             {1,1,1}    {3,3}          {3,4}
             {1,1,1,1}  {1,1,2}        {4,4}
                        {1,1,1,2}      {1,1,3}
                        {1,1,1,1,1}    {1,2,2}
                        {1,1,1,1,1,1}  {2,2,2}
                                       {1,1,1,3}
                                       {1,1,2,2}
                                       {1,2,2,2}
                                       {2,2,2,2}
                                       {1,1,1,1,2}
                                       {1,1,1,1,1,2}
                                       {1,1,1,1,1,1,1}
                                       {1,1,1,1,1,1,1,1}
For example, the multiset y = {1,1,1,1,2} has right half (inclusive) {1,1,2}, with sum 4, so y is counted under a(4).

Links

Crossrefs

The exclusive version is A360673.
Column sums of A360675 with rows reversed.
The case of sets is A360955, exclusive A360954.
The even-length case is A360956.
A360672 counts partitions by left sum (exclusive).
A360679 gives right sum (inclusive) of prime indices.
Cf. A000041, A347044, A361200, A361201.

Programs

  • Mathematica
    Table[Length[Select[Join@@IntegerPartitions/@Range[0,3*k], Total[Take[#,Ceiling[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*(2-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*(2 - x^k)/((1 - x^k)^(k+1) * Product_{j=1..k-1} (1-x^j)). - Andrew Howroyd, Mar 11 2023

Extensions

Terms a(24) and beyond from Andrew Howroyd, Mar 11 2023

A360673 Number of multisets of positive integers whose right half (exclusive) sums to n.

Original entry on oeis.org

1, 2, 7, 13, 27, 37, 73, 89, 156, 205, 315, 387, 644, 749, 1104, 1442, 2015, 2453, 3529, 4239, 5926, 7360, 9624, 11842, 16115, 19445, 25084, 31137, 39911, 48374, 62559, 75135, 95263, 115763, 143749, 174874, 218614, 261419, 321991, 388712, 477439, 569968, 698493
Offset: 0

Views

Author

Gus Wiseman, Mar 04 2023

Keywords

Examples

			The a(0) = 1 through a(3) = 13 multisets:
  {}  {1,1}    {1,2}        {1,3}
      {1,1,1}  {2,2}        {2,3}
               {1,1,2}      {3,3}
               {1,2,2}      {1,1,3}
               {2,2,2}      {1,2,3}
               {1,1,1,1}    {1,3,3}
               {1,1,1,1,1}  {2,2,3}
                            {2,3,3}
                            {3,3,3}
                            {1,1,1,2}
                            {1,1,1,1,2}
                            {1,1,1,1,1,1}
                            {1,1,1,1,1,1,1}
For example, the multiset y = {1,1,1,1,2} has right half (exclusive) {1,2}, with sum 3, so y is counted under a(3).

Links

Crossrefs

The inclusive version is A360671.
Column sums of A360672.
The case of sets is A360954, inclusive A360955.
The even-length case is A360956.
A359893 and A359901 count partitions by median.
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, A360616, A360617, A360674, A360675, A360953.

Programs

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

Formula

G.f.: 1 + Sum_{k>=1} x^k*(2 - x^k)/((1 - x^k)^(k+2) * Product_{j=1..k-1} (1-x^j)). - Andrew Howroyd, Mar 11 2023

Extensions

Terms a(21) and beyond from Andrew Howroyd, Mar 11 2023

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

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
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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