A117488 -id:A117488 - cp's OEIS frontend

A117489 Row sums of A117488 which has 1, 3, 5, 7, ... entries per row.

1, 6, 30, 115, 374, 1079, 2848, 7005, 16285, 36117, 76981, 158563, 317014, 617349, 1174400, 2187584, 3998057, 7181362, 12696179, 22120549, 38023748, 64546190, 108296742, 179729895, 295244362, 480358370, 774483006, 1238052200

Offset: 1

Alford Arnold, Mar 22 2006

nonn

Note that the central diagonal of table A117488 is 1 2 5 20 ...

			A117488 begins
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
so a(n) begins
1 6 30 115 374 ...

Cf. A115994, A117488.

More terms from David Wasserman, Nov 03 2010

A117566 Strip A117488 of r-1 row values 1,2,5,10,20,36,65 ... A000712.

Original entry on oeis.org

1, 2, 3, 5, 8, 14, 10, 18, 30, 49, 20, 34, 59, 94, 149, 36, 63, 104, 169, 264, 405, 65, 108, 179, 284, 445, 676, 1017, 110, 183, 294, 465, 716, 1089, 1622, 2387, 185, 298, 475, 736, 1129, 1694, 2517, 3678, 5324, 300, 479, 746, 1149, 1734, 2589, 3808, 5544, 7972

Offset: 1

Alford Arnold, Mar 29 2006

Note that the Durfee Square sequences addressed in A115994, A117488 and this triangular array can be generated using self-convolution of diagonals in the partition array A008284.

			Row four of A117488 is 1 2 5 10 18 30 49
remove 3 terms 1 2 5 so
row four of A117566 is 10 18 30 49

Cf. A000041 A008284 A115994 A117488.

T(n,k) = A117488(n,n+k-1), 1<=k<=n . - R. J. Mathar, Jan 22 2008

More terms from R. J. Mathar, Jan 22 2008

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

Alford Arnold, Mar 22 2006

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)

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

Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-3,0,6,0,-3,-2,1,2,-1).
Index entries for two-way infinite sequences

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

n:=3; G:=SymmetricGroup(n); H:=DirectProduct(G,G); MolienSeries(H); // N. J. A. Sloane, Mar 10 2007

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

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 *)

Vec(x^9 / ((1 - x)^6*(1 + x)^2*(1 + x + x^2)^2) + O(x^60)) \\ Colin Barker, Dec 12 2019

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

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)

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

Alford Arnold, Mar 22 2006

nonn

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 ).

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2, 1, -2, -1, -2, 0, 2, 6, 2, -3, -6, -5, -2, 3, 12, 3, -2, -5, -6, -3, 2, 6, 2, 0, -2, -1, -2, 1, 2, -1).

Cf. A000027, A000712, A001399, A001400, A006918.

Cf. A115994, A117485, A117486, A117488, A117489.

n:=5; G:=SymmetricGroup(n); H:=DirectProduct(G,G); MolienSeries(H); // N. J. A. Sloane

# 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

CoefficientList[Series[1/(Product[(1-x^j), {j,5}])^2, {x,0,45}], x] (* G. C. Greubel, Jan 01 2020 *)

my(x='x+O('x^45)); Vec( 1/(prod(j=1,5, 1-x^j))^2 ) \\ G. C. Greubel, Jan 01 2020

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

More terms from R. J. Mathar, Apr 07 2006

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

cp's OEIS Frontend

A117489 Row sums of A117488 which has 1, 3, 5, 7, ... entries per row.

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A117566 Strip A117488 of r-1 row values 1,2,5,10,20,36,65 ... A000712.

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

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Magma

Maple

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

Views

Author

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Comments

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Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Extensions

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

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