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.

A072213 Number of partitions of n^2.

Original entry on oeis.org

1, 1, 5, 30, 231, 1958, 17977, 173525, 1741630, 18004327, 190569292, 2056148051, 22540654445, 250438925115, 2814570987591, 31946390696157, 365749566870782, 4219388528587095, 49005643635237875, 572612058898037559
Offset: 0

Views

Author

Jeff Burch, Jul 03 2002

Keywords

Links

Crossrefs

Cf. A000041, A000290, A072243, A219430.
Cf. A161407, A161408. - Reinhard Zumkeller, Jun 10 2009

Programs

  • Maple
    A072213 := proc(n) combinat[numbpart](n^2) ; end proc:
seq(A072213(n),n=0..10) ; # R. J. Mathar, Jan 24 2011
  • Mathematica
    Table[ PartitionsP[n^2], {n, 1, 20}]
  • PARI
    a(n)=numbpart(n^2)
  • PARI
    a(n)=polcoeff(1/eta(x),n^2,x)
  • Sage
    [number_of_partitions(n^2)for n in range(0,26)] # Zerinvary Lajos, Nov 26 2009

Formula

a(n) = A000041(n^2).
a(n) ~ exp(Pi*sqrt(2/3)*n) / (4*sqrt(3)*n^2). - Vaclav Kotesovec, Dec 01 2015

A093116 Number of partitions of n^2 into squares not less than n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 1, 2, 5, 4, 4, 5, 9, 15, 23, 24, 13, 20, 32, 55, 84, 113, 185, 303, 545, 167, 298, 435, 716, 1055, 1701, 2584, 4213, 6471, 10218, 15884, 4856, 7376, 11231, 17221, 26054, 39583, 60109, 91622, 138569, 209951, 318368, 483098, 730183
Offset: 0

Views

Author

Reinhard Zumkeller, Mar 21 2004

Keywords

Examples

			n=10: 10^2 = 100 = 64+36 = 36+16+16+16+16 = 25+25+25+25, all other partitions of 100 into squares contain parts < 10, therefore a(10) = 4.

Links

Crossrefs

Cf. A093115, A092362, A001156, A037444, A078134.
Cf. A072213, A161408. [Reinhard Zumkeller, Jun 10 2009]

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i^2>n, 0, b(n, i+1) +b(n-i^2, i)))
    end:
a:= proc(n) local r; r:= isqrt(n);
      b(n^2, r+`if`(r^2Alois P. Heinz, Apr 15 2013
  • Mathematica
    b[n_, i_] := b[n, i] = If[n==0, 1, If[i^2>n, 0, b[n, i+1] + b[n-i^2, i]]]; a[n_] := With[{r = Sqrt[n]//Floor}, b[n^2, r + If[r^2Jean-François Alcover, Oct 26 2015, after Alois P. Heinz *)

A161407 Number of partitions of n^2 into parts smaller than n.

Original entry on oeis.org

1, 0, 1, 5, 30, 185, 1226, 8442, 60289, 442089, 3314203, 25295011, 195990980, 1538069121, 12203218743, 97746332667, 789480879664, 6423539487002, 52607252796831, 433368610079872, 3588859890833443, 29862449600982149, 249560820679038935, 2093852201126089073
Offset: 0

Views

Author

Reinhard Zumkeller, Jun 10 2009

Keywords

Examples

			a(3) = #{2+2+2+2+1, 2+2+2+1+1+1, 2+2+5x1, 2+7x1, 9x1} = 5.

Links

Crossrefs

Cf. A072213, A093115, A109655, A161408.

Programs

  • Maple
    a := proc (n) local G, Gser: G := 1/(product(1-x^j, j = 1 .. n-1)): Gser := series(G, x = 0, n^2+5): coeff(Gser, x, n^2) end proc: 1, seq(a(n), n = 1 .. 23); # Emeric Deutsch, Jun 20 2009
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
    end:
a:= n-> b(n^2, n-1):
seq(a(n), n=0..30);  # Alois P. Heinz, Dec 21 2014
  • Mathematica
    b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]]]; a[n_] := b[n^2, n-1]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 30 2015, after Alois P. Heinz *)

Formula

a(n) ~ c * d^n / n^2, where d = A258268 = 9.153370192454122461948530292401354... and c = 0.0881548837986971165169272782933415... - Vaclav Kotesovec, Sep 08 2021

Extensions

More terms from Emeric Deutsch, Jun 20 2009
a(0)=1 from Alois P. Heinz, Dec 21 2014

A347585 Number of partitions of n^2 into n or more parts.

Original entry on oeis.org

1, 1, 4, 25, 201, 1773, 16751, 165083, 1681341, 17562238, 187255089, 2030853040, 22344663465, 248900855994, 2802367768848, 31848644363490, 364960085991118, 4212964989100093, 48953036382441044, 572178690287957687, 6723501191850208483, 79388206896842420091
Offset: 0

Views

Author

Seiichi Manyama, Sep 08 2021

Keywords

Links

Crossrefs

Cf. A058399, A072213, A161408, A206226, A206240.

Formula

a(n) = [x^(n^2)] Sum_{k>=n} x^k / Product_{j=1..k} (1 - x^j).
a(n) = A072213(n) + A206240(n) - A206226(n).
a(n) ~ exp(Pi*sqrt(2/3)*n) / (4*sqrt(3)*n^2). - Vaclav Kotesovec, Sep 14 2021
Showing 1-4 of 4 results.

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

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