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 1-3 of 3 results.

A117485 Expansion of x^9/((1-x)*(1-x^2)*(1-x^3))^2.

Original entry on oeis.org

1, 2, 5, 10, 18, 30, 49, 74, 110, 158, 221, 302, 407, 536, 698, 896, 1136, 1424, 1770, 2176, 2656, 3216, 3866, 4616, 5481, 6466, 7591, 8866, 10306, 11926, 13747, 15778, 18046, 20566, 23359, 26446, 29855, 33600, 37716, 42224, 47152, 52528, 58388, 64752, 71664
Offset: 9

Views

Author

Alford Arnold, Mar 22 2006

Keywords

Comments

Molien series for S_3 X S_3, cf. A001399.
From Gus Wiseman, Apr 06 2019: (Start)
Also the number of integer partitions of n with Durfee square of length 3. The Heinz numbers of these partitions are given by A307386. For example, the a(9) = 1 through a(13) = 18 partitions are:
(333) (433) (443) (444) (544)
(3331) (533) (543) (553)
(3332) (633) (643)
(4331) (3333) (733)
(33311) (4332) (4333)
(4431) (4432)
(5331) (4441)
(33321) (5332)
(43311) (5431)
(333111) (6331)
(33322)
(33331)
(43321)
(44311)
(53311)
(333211)
(433111)
(3331111)
(End)

Examples

			As a cross-check, row sixteen of A115994 yields p(16) = 16 + 140 + 74 + 1.

Links

Crossrefs

Column k=3 of A115994.
Cf. A000027 (for k=1), A006918 (for k=2), A117488, A117489, A001399, A117486.
Cf. A115720, A307386, A325164.

Programs

  • Magma
    n:=3; G:=SymmetricGroup(n); H:=DirectProduct(G,G); MolienSeries(H); // N. J. A. Sloane, Mar 10 2007
  • Maple
    with(combstruct):ZL:=[st, {st=Prod(left, right), left=Set(U, card=r), right=Set(U, card=r), U=Sequence(Z, card>=1)}, unlabeled]: subs(r=3, stack): seq(count(subs(r=3, ZL), size=m), m=6..50) ; # Zerinvary Lajos, Jan 02 2008
  • Mathematica
    CoefficientList[Series[1/((1-x)(1-x^2)(1-x^3))^2,{x,0,50}],x] (* Harvey P. Dale, Oct 09 2011 *)
durf[ptn_]:=Length[Select[Range[Length[ptn]],ptn[[#]]>=#&]];
Table[Length[Select[IntegerPartitions[n],durf[#]==3&]],{n,0,30}] (* Gus Wiseman, Apr 06 2019 *)
  • PARI
    Vec(x^9 / ((1 - x)^6*(1 + x)^2*(1 + x + x^2)^2) + O(x^60)) \\ Colin Barker, Dec 12 2019
  • PARI
    a(n) = floor((3*n^5 - 45*n^4 + 200*n^3 - 180*n^2 - 363*n + 1600)/12960 + n/27*(n%3==0) - n/32*(n%2==0)) \\ Hoang Xuan Thanh, Jul 17 2025

Formula

a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - 3*a(n-4) + 6*a(n-6) - 3*a(n-8) - 2*a(n-9) + a(n-10) + 2*a(n-11) - a(n-12) for n>20. - Colin Barker, Dec 12 2019
From Hoang Xuan Thanh, May 17 2025: (Start)
a(n+3) = Sum_{x+2*y+3*z=n} x*y*z.
a(n+3) = n*(n^2-1)*(3*n^2-67)/12960 - floor((n+1)/3)/27 + [n mod 2 = 0]*n/32 + [n mod 3 = 0]*n/27 where [] is the Iverson bracket. (End)

Extensions

Entry revised by N. J. A. Sloane, Mar 10 2007

A117487 G.f.: 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5))^2.

Original entry on oeis.org

1, 2, 5, 10, 20, 36, 63, 104, 169, 264, 405, 604, 888, 1278, 1815, 2536, 3502, 4772, 6437, 8586, 11352, 14866, 19315, 24890, 31851, 40466, 51089, 64092, 79952, 99172, 122386, 150264, 183639, 223394, 270605, 326422, 392225, 469490, 559970, 665542, 788412
Offset: 1

Views

Author

Alford Arnold, Mar 22 2006

Keywords

Comments

Molien series for S_5 X S_5, cf. A001401.
Molien series for S_k X S_k approaches A000712 as k increases.
Column 5 of table A115994.
Note that a(5) is 20, the scalar product of (1 1 2 3 5) and (5 3 2 1 1 ). a(6) is 36, the scalar product of (1 1 2 3 5 7) and (7 5 3 2 1 1 ).

Links

Crossrefs

Cf. A000027, A000712, A001399, A001400, A006918.
Cf. A115994, A117485, A117486, A117488, A117489.

Programs

  • Magma
    n:=5; G:=SymmetricGroup(n); H:=DirectProduct(G,G); MolienSeries(H); // N. J. A. Sloane
  • Maple
    # adapted from A115994 kmax := 120 : qmax := kmax/2 : g:=sum(t^k*q^(k^2)/product((1-q^j)^2, j=1..k), k=1..kmax): gser:=series(g, q=0, qmax): for n from 25 to qmax-1 do P :=coeff(gser, q^n) : printf("%a,",coeff(P, t^5)); od: # R. J. Mathar, Apr 07 2006
  • Mathematica
    CoefficientList[Series[1/(Product[(1-x^j), {j,5}])^2, {x,0,45}], x] (* G. C. Greubel, Jan 01 2020 *)
  • PARI
    my(x='x+O('x^45)); Vec( 1/(prod(j=1,5, 1-x^j))^2 ) \\ G. C. Greubel, Jan 01 2020
  • Sage
    def A117487_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(product(1-x^j for j in (1..5)))^2 ).list()
A117487_list(45) # G. C. Greubel, Jan 01 2020

Extensions

More terms from R. J. Mathar, Apr 07 2006
Entry revised by N. J. A. Sloane, Mar 10 2007

A117488 The number 1 followed by 2*k+1 terms from column k of table A115994.

Original entry on oeis.org

1, 1, 2, 3, 1, 2, 5, 8, 14, 1, 2, 5, 10, 18, 30, 49, 1, 2, 5, 10, 20, 34, 59, 94, 149, 1, 2, 5, 10, 20, 36, 63, 104, 169, 264, 405, 1, 2, 5, 10, 20, 36, 65, 108, 179, 284, 445, 676, 1017, 1, 2, 5, 10, 20, 36, 65, 110, 183, 294, 465, 716, 1089, 1622, 2387, 1, 2, 5, 10, 20, 36, 65
Offset: 1

Views

Author

Alford Arnold, Mar 22 2006

Keywords

Examples

			Column two of table A115994 begins 1 2 5 8 14 20 30 ... A006918
so row three of A117488 is 1 2 5 8 14.
Triangle starts
1
1 2 3
1 2 5 8 14
1 2 5 10 18 30 49
1 2 5 10 20 34 59 94 149
1 2 5 10 20 36 63 104 169 264 405
1 2 5 10 20 36 65 108 179 284 445 676 1017
1 2 5 10 20 36 65 110 183 294 465 716 1089 1622 2387

Crossrefs

Cf. A006918, A115994, A117489.

Programs

  • Maple
    A026820 := proc(n,k) if k > n then combinat[numbpart](n,n) ; else combinat[numbpart](n,k) ; fi ; end: A115994 := proc(n,k) local i ; add(A026820(i,k)*A026820(n-k^2-i,k),i=0..n-k^2) ; end: A117488 := proc(n,k) if k >= 2*n then 0 ; else if n = 1 then 1; else A115994(k+n^2-2*n,n-1) ; fi ; fi ; end: for n from 1 to 10 do for k from 1 to 2*n-1 do printf("%d ",A117488(n,k)) ; od ; od ; # R. J. Mathar, Feb 22 2007

Extensions

More terms from R. J. Mathar, Feb 22 2007
Showing 1-3 of 3 results.

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

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