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

A093115 Number of partitions of n^2 into squares not greater than n.

Original entry on oeis.org

1, 1, 1, 1, 5, 7, 10, 13, 17, 108, 159, 228, 317, 430, 572, 748, 5753, 8125, 11266, 15376, 20672, 27430, 35942, 46575, 59717, 523905, 708028, 946875, 1253880, 1645224, 2140099, 2761318, 3535658, 4494602, 5674753, 7118724, 69766770, 90940578, 117756370
Offset: 0

Views

Author

Reinhard Zumkeller, Mar 21 2004

Keywords

Examples

			n=6: 6^2 = 9*2^2 = 8*2^2+4*1^2 = 7*2^2+8*1^2 = 6*2^2+12*1^2 = 5*2^2+16*1^2 = 4*2^2+20*1^2 = 3*2^2+24*1^2 = 2*2^2+28*1^2 = 1*2^2+32*1^2 = 36*1^2, therefore a(6)=10.

Links

Crossrefs

Cf. A093116, A092362, A001156, A037444, A078134.
Cf. A072925.
Cf. A072213, A161407. [Reinhard Zumkeller, Jun 10 2009]

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i<1, 0, b(n, i-1) +`if`(i^2>n, 0, b(n-i^2, i))))
    end:
a:= proc(n) local r; r:= isqrt(n);
      b(n^2, r-`if`(r^2>n, 1, 0))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 15 2013
  • Mathematica
    b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i^2 > n, 0, b[n-i^2, i]]]]; a[n_] := (r = Sqrt[n] // Floor; b[n^2, r - If[r^2 > n, 1, 0]]); Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 29 2015, after Alois P. Heinz *)

Formula

Coefficient of x^(n^2) in the series expansion of Product_{k=1..floor(sqrt(n))} 1/(1 - x^(k^2)). - Vladeta Jovovic, Mar 24 2004

Extensions

More terms from Vladeta Jovovic, Mar 24 2004
Corrected a(0) by Alois P. Heinz, Apr 15 2013

A109655 Number of partitions of n^2 into up to n parts each no more than 2n, or of n(3n+1)/2 into exactly n distinct parts each no more than 3n.

Original entry on oeis.org

1, 1, 3, 8, 33, 141, 676, 3370, 17575, 94257, 517971, 2900900, 16509188, 95220378, 555546058, 3273480400, 19456066175, 116521302221, 702567455381, 4261765991164, 25992285913221, 159303547578873, 980701254662294, 6061894625462492, 37609015174472628
Offset: 0

Views

Author

Henry Bottomley, Aug 05 2005

Keywords

Examples

			a(3) = 8 since 3^2=9 can be partitioned into 3+3+3, 4+3+2, 4+4+1, 5+4, 5+3+1, 5+2+2, 6+3, or 6+2+1, while 3*(3*3+1)/2=15 can be partitioned into 6+5+4, 7+5+3, 7+6+2, 8+6+1, 8+5+2, 8+4+3, 9+5+1, or 9+4+2.

Links

Crossrefs

Cf. A161407. - Reinhard Zumkeller, Jun 10 2009
Row n=3 of A204459. - Alois P. Heinz, Jan 18 2012

Programs

  • Maple
    b:= proc(n, i, t) option remember;
      `if`(it*(2*i-t+1)/2, 0,
      `if`(n=0, 1, b(n, i-1, t) +`if`(n b(n*(3*n+1)/2, 3*n, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Jan 18 2012
  • Mathematica
    b[n_, i_, t_] := b[n, i, t] = If[it*(2*i-t+1)/2, 0, If[n == 0, 1, b[n, i-1, t] + If[nJean-François Alcover, Oct 05 2015, after Alois P. Heinz *)

Formula

a(n) = A067059(n,2n) = A067059(2n,n).
Slightly less than but close to (27/4)^n*sqrt(3)/(2*Pi*n^2).

Extensions

More terms from Alois P. Heinz, Jan 18 2012

A161408 Number of partitions of n^2 into parts greater than n.

Original entry on oeis.org

1, 0, 1, 2, 6, 17, 56, 179, 607, 2076, 7269, 25736, 92360, 334506, 1222463, 4499892, 16673446, 62130710, 232700539, 875483029, 3307244256, 12539455600, 47702381244, 182021195608, 696487788847, 2671877845634, 10274126646175, 39593295985708, 152889766657797
Offset: 0

Views

Author

Reinhard Zumkeller, Jun 10 2009

Keywords

Examples

			a(4) = #{16, 11+5, 10+6, 9+7, 8+8, 6+5+5} = 6.

Links

Crossrefs

Cf. A072213, A093116, A161407.

Programs

  • Maple
    a := proc (n) local G, Gser: G := 1/(product(1-x^j, j = n+1 .. n^2)): Gser := series(G, x = 0, n^2+5): coeff(Gser, x, n^2) end proc: 1, seq(a(n), n = 1 .. 27); # Emeric Deutsch, Jun 22 2009
  • Mathematica
    a[n_] := a[n] = 1/Product[1 - x^j, {j, n + 1, n^2}] + O[x]^(n^2 + 1) // CoefficientList[#, x]& // Last;
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 28}] (* Jean-François Alcover, May 18 2017, adapted from Maple *)

Formula

a(n) ~ c * d^n / n^(3/2), where d = 4.076293... and c = 0.187307... - Vaclav Kotesovec, Sep 08 2021

Extensions

Extended by Emeric Deutsch, Jun 22 2009
a(0)=1 from Alois P. Heinz, Dec 21 2014
Showing 1-4 of 4 results.

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