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.

A206226 Number of partitions of n^2 into parts not greater than n.

Original entry on oeis.org

1, 1, 3, 12, 64, 377, 2432, 16475, 116263, 845105, 6292069, 47759392, 368379006, 2879998966, 22777018771, 181938716422, 1465972415692, 11902724768574, 97299665768397, 800212617435074, 6617003142869419, 54985826573015541, 458962108485797208, 3846526994743330075
Offset: 0

Views

Author

Paul D. Hanna, Feb 05 2012

Keywords

Comments

Also the number of partitions of n^2 using n or fewer numbers. Thus for n=3 one has: 9; 1,8; 2,7; 3,6; 4,5; 1,1,7; 1,2,6; 1,3,5; 1,4,4; 2,2,5; 2,3,4; 3,3,3. - J. M. Bergot, Mar 26 2014 [computations done by Charles R Greathouse IV]
The partitions in the comments above are the conjugates of the partitions in the definition. By conjugation we have: "partitions into parts <= m" are equinumerous with "partitions into at most m parts". - Joerg Arndt, Mar 31 2014
From Vaclav Kotesovec, May 25 2015: (Start)
In general, "number of partitions of j*n^2 into parts that are at most n" is (for j>0) asymptotic to c(j) * d(j)^n / n^2, where c(j) and d(j) are a constants.
-------
j c(j)
1 0.1582087202672504149766310999238...
2 0.0794245035465730707705885572860...
3 0.0530017980244665552354063060738...
4 0.0397666338404544208556554596295...
5 0.0318193213988281353709268311928...
...
17 0.0093617308583114626385718275875...
c(j) for big j asymptotically approaches 1 / (2*Pi*j).
---------
j d(j)
1 9.15337019245412246194853029240... = A258268
2 16.57962120993269533568313969522...
3 23.98280768122086592445663786762...
4 31.37931997386325137074644287711...
5 38.77298550971449870728474612568...
...
17 127.45526806942537991146993713837...
d(j) for big j asymptotically approaches j * exp(2).
(End)
d(j) = r^(2*j+1)/(r-1), where r is the root of the equation polylog(2, 1-r) + (j+1/2)*log(r)^2 = 0. - Vaclav Kotesovec, Jun 11 2015

Links

Crossrefs

Cf. A173519, A206227, A206240, A107379, A258268.
Column k=2 of A238016.
Cf. A258296 (j=2), A258293 (j=3), A258294 (j=4), A258295 (j=5).

Programs

  • Maple
    T:= proc(n, k) option remember;
      `if`(n=0 or k=1, 1, T(n, k-1) + `if`(k>n, 0, T(n-k, k)))
    end:
seq(T(n^2, n), n=0..20); # Vaclav Kotesovec, May 25 2015 after Alois P. Heinz
  • Mathematica
    Table[SeriesCoefficient[Product[1/(1-x^k),{k,1,n}],{x,0,n^2}],{n,0,20}] (* Vaclav Kotesovec, May 25 2015 *)
(* A program to compute the constants d(j) *) Table[r^(2*j+1)/(r-1) /.FindRoot[-PolyLog[2,1-r] == (j+1/2)*Log[r]^2, {r, E}, WorkingPrecision->60], {j, 1, 5}] (* Vaclav Kotesovec, Jun 11 2015 *)
  • PARI
    {a(n)=polcoeff(prod(k=1,n,1/(1-x^k+x*O(x^(n^2)))),n^2)}
for(n=0,25,print1(a(n),", "))

Formula

a(n) = [x^(n^2)] Product_{k=1..n} 1/(1 - x^k).
a(n) ~ c * d^n / n^2, where d = 9.1533701924541224619485302924013545... = A258268, c = 0.1582087202672504149766310999238742... . - Vaclav Kotesovec, Sep 07 2014

A258294 Number of partitions of 4*n^2 into parts that are at most n.

Original entry on oeis.org

1, 1, 9, 127, 2280, 46262, 1015691, 23541165, 567852809, 14123231487, 359874480333, 9351900623083, 247006639629275, 6613877399621729, 179171447281396640, 4902895256737984134, 135346525073067516814, 3765244155890019687101, 105465364199865165010867
Offset: 0

Views

Author

Vaclav Kotesovec, May 25 2015

Keywords

Links

Crossrefs

Cf. A206226, A258296, A258293, A258295.

Programs

  • Maple
    T:=proc(n, k) option remember; `if`(n=0 or k=1, 1, T(n, k-1) + `if`(n
  • Mathematica
    (* A program to compute the constant d = 31.37931997... *) With[{j=4}, r^(2*j+1)/(r-1) /.FindRoot[-PolyLog[2,1-r] == (j+1/2)*Log[r]^2, {r, E}, WorkingPrecision->100]] (* Vaclav Kotesovec, Jun 10 2015 *)

Formula

a(n) ~ c * d^n / n^2, where d = 31.379319973863251370746442877119704410889..., c = 0.0397666338404544208556554596295683858... .

A258295 Number of partitions of 5*n^2 into parts that are at most n.

Original entry on oeis.org

1, 1, 11, 192, 4263, 106852, 2897747, 82966258, 2472338185, 75966810293, 2391508958235, 76782438832425, 2505642670439980, 82893573492724961, 2774547946438608789, 93807671621922558215, 3199617653993448321146, 109979504522862990517172, 3806257106793028952525938
Offset: 0

Views

Author

Vaclav Kotesovec, May 25 2015

Keywords

Links

Crossrefs

Cf. A206226, A258296, A258293, A258294.

Programs

  • Maple
    T:=proc(n, k) option remember; `if`(n=0 or k=1, 1, T(n, k-1) + `if`(n
  • Mathematica
    (* A program to compute the constant d = 38.7729855... *) With[{j=5}, r^(2*j+1)/(r-1) /.FindRoot[-PolyLog[2,1-r] == (j+1/2)*Log[r]^2, {r, E}, WorkingPrecision->100]] (* Vaclav Kotesovec, Jun 10 2015 *)

Formula

a(n) ~ c * d^n / n^2, where d = 38.7729855097144987072847461256815071909..., c = 0.0318193213988281353709268311928... .

A258296 Number of partitions of 2*n^2 into parts that are at most n.

Original entry on oeis.org

1, 1, 5, 37, 351, 3765, 43752, 536375, 6842599, 89984614, 1212199424, 16651935901, 232477235048, 3290090540717, 47106320777132, 681247106742555, 9938641464083052, 146113228303254020, 2162784490438698636, 32209221982817148364, 482304350308369699381
Offset: 0

Views

Author

Vaclav Kotesovec, May 25 2015

Keywords

Links

Crossrefs

Cf. A206226, A258293, A258294, A258295.

Programs

  • Maple
    T:=proc(n,k) option remember; `if`(n=0 or k=1, 1, T(n,k-1) + `if`(n
  • Mathematica
    (* A program to compute the constant d = 16.5796212... *) With[{j=2}, r^(2*j+1)/(r-1) /.FindRoot[-PolyLog[2,1-r] == (j+1/2)*Log[r]^2, {r, E}, WorkingPrecision->100]] (* Vaclav Kotesovec, Jun 10 2015 *)

Formula

a(n) ~ c * d^n / n^2, where d = 16.57962120993269533568313969522872808998..., c = 0.07942450354657307077058855728600800998... .
Showing 1-4 of 4 results.

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