A002625 -id:A002625 - cp's OEIS frontend

A097701 Expansion of 1/((1-x)^2(1-x^2)^2(1-x^3)).

1, 2, 5, 9, 16, 25, 39, 56, 80, 109, 147, 192, 249, 315, 396, 489, 600, 726, 874, 1040, 1232, 1446, 1690, 1960, 2265, 2600, 2975, 3385, 3840, 4335, 4881, 5472, 6120, 6819, 7581, 8400, 9289, 10241, 11270, 12369, 13552, 14812, 16164, 17600, 19136

Offset: 0

Ralf Stephan, Aug 24 2004

Number of partitions of 5*n+12 or 5*n+13 into 5 parts (+-) 3 mod 5. For example, the a(3) = 9 partitions of 27 are: [18,3,2,2,2], [13,8,2,2,2], [17,3,3,2,2], [12,7,3,3,2], [7,7,7,3,3], [13,7,3,2,2], [8,8,7,2,2], [12,8,3,2,2], [8,7,7,3,2]. - Richard Turk, Apr 23 2016

Number of partitions of n into two kinds of parts 1, two kinds of parts 2, and one kind of parts 3. - Joerg Arndt, Apr 24 2016

			G.f. = 1 + 2*x + 5*x^2 + 9*x^3 + 16*x^4 + 25*x^5 + 39*x^6 + ... - _Michael Somos_, Aug 16 2023

Index entries for linear recurrences with constant coefficients, signature (2,1,-3,-1,1,3,-1,-2,1).

First differences of A002625. Partial sums of A008763.

with(combstruct):ZL:=[st,{st=Prod(left,right),left=Set(U,card=r),right=Set(U,card=1)}, unlabeled]: subs(r=5,stack): seq(count(subs(r=3,ZL),size=m),m=3..47) ; # Zerinvary Lajos, Mar 09 2007

CoefficientList[Series[1/((1-x)^2(1-x^2)^2(1-x^3)),{x,0,50}],x] (* or *) LinearRecurrence[{2,1,-3,-1,1,3,-1,-2,1},{1,2,5,9,16,25,39,56,80},50] (* Harvey P. Dale, May 20 2013 *)
a[ n_] := Round[(n + 1)*(9*(-1)^n + n^3 + 17*n^2 + 95*n + 184)/288]; (* Michael Somos, Aug 16 2023*)

a(n)=1/576*(2*n^4+36*n^3+224*n^2+558*n+495+(18*n+81)*(-1)^n-64*(if(n%3,1,0)))

x='x+O('x^99); Vec(1/((1-x)^2*(1-x^2)^2*(1-x^3))) \\ Altug Alkan, Sep 18 2016

a(n) = floor((n + 1) * (9*(-1)^n + n^3 + 17*n^2 + 95*n + 184)/288 + 1/2). - Tani Akinari, Oct 07 2012
a(n) = 2*a(n-1) + a(n-2) - 3*a(n-3) - a(n-4) + a(n-5) + 3*a(n-6) - a(n-7) - 2*a(n-8) + a(n-9) for n >= 9, with initial values as shown. - Harvey P. Dale, May 20 2013
a(n) = (6*n*(9*((-1)^n + 31) + n*(n*(n + 18) + 112)) + 243*(-1)^n + 128*cos((2*Pi*n)/3) + 1357)/1728. - Ilya Gutkovskiy, Apr 23 2016
a(n) = 1 + 175*n/288 + 47*n^2/144 + n^3/16 + n^4/288 + (9/16 + n/8)*floor(n/2) + 2*floor(n/3)/9 + floor((n+1)/3)/9. - Vaclav Kotesovec, Apr 24 2016
a(n) = a(-9-n) for all n in Z. - Michael Somos, Aug 16 2023

A164680 Expansion of x/((1-x)^3(1-x^2)^3(1-x^3)).

Original entry on oeis.org

1, 3, 9, 20, 42, 78, 139, 231, 372, 573, 861, 1254, 1791, 2499, 3432, 4629, 6162, 8085, 10492, 13455, 17094, 21503, 26832, 33201, 40795, 49764, 60333, 72687, 87096, 103785, 123075, 145236, 170646, 199626, 232617, 269997, 312277, 359898, 413448, 473438

Offset: 1

Alford Arnold, Aug 21 2009

Convolution of A006918 with A001399, or of A002625 with A059841 (A000035 if offsets are respected),
or of A038163 with A022003 or of A057524 with A027656 or of A014125 with the aerated version of A000217,
or of A002624 with A103221, or of A002623 with A008731, or of other combinations of splitting the signature -/3,3,1 into two components.
If we apply the enumeration of Molien series as described in A139672,
this is row 45=9*5 of a table of values related to Molien series, i.e., the
product of the sequence on row 9 (A006918) with the sequence on row 5 (A001399).
This is associated with the root system E6, and can be described using the additive function on the affine E6 diagram:
      1
      |
      2
      |
1--2--3--2--1

			To calculate a(3), we consider the first three terms of A001399 = (1 1 2...)
and the first three terms of A006918 = (1 2 5 ...), to get the convolved a(3) = 1*5+1*2+2*1 = 9.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (3,0,-7,3,6,0,-6,-3,7,0,-3,1).

Cf. A139672 (row 21).
For G2, the corresponding sequence is A001399.
For F4, the corresponding sequence is A115264.
For E7, the corresponding sequence is A210068.
For E8, the corresponding sequence is A045513.
See A210634 for a closely related sequence.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/((1-x)^3*(1-x^2)^3*(1-x^3)) )); // G. C. Greubel, Jan 13 2020

seq(coeff(series(x/((1-x)^3*(1-x^2)^3*(1-x^3)), x, n+1), x, n), n = 1..40); # G. C. Greubel, Jan 13 2020

Rest@CoefficientList[Series[x/((1-x)^3*(1-x^2)^3*(1-x^3)), {x,0,40}], x] (* G. C. Greubel, Jan 13 2020 *)
LinearRecurrence[{3,0,-7,3,6,0,-6,-3,7,0,-3,1},{1,3,9,20,42,78,139,231,372,573,861,1254},40] (* Harvey P. Dale, Aug 03 2025 *)

Vec(1/(1-x)^3/(1-x^2)^3/(1-x^3)+O(x^40)) \\ Charles R Greathouse IV, Sep 23 2012

x=PowerSeriesRing(QQ, 'x', 40).gen()
1/((1-x)^3*(1-x^2)^3*(1-x^3))

a(n) = round( -(-1)^n*(n+3)*(n+7)/256 +(6*n^6 +180*n^5 +2070*n^4 +11400*n^3 +30429*n^2 +34290*n +9785)/103680 ) - R. J. Mathar, Mar 19 2012

Edited and extended by R. J. Mathar, Aug 22 2009

Corrected link to index entries - R. J. Mathar, Aug 26 2009

A206228 a(n) = [x^n] Product_{k=1..n} 1/(1 - x^k)^(n-k+1).

Original entry on oeis.org

1, 1, 4, 17, 80, 384, 1887, 9385, 47139, 238488, 1213588, 6204547, 31844710, 163978344, 846741721, 4382945317, 22735196277, 118151632006, 615032941924, 3206257881171, 16736910271178, 87472908459696, 457662760258109, 2396899780970552, 12564645719730297

Offset: 0

Paul D. Hanna, Feb 05 2012

nonn

Number of partitions of n with 1 kind of n's, 2 kinds of (n-1)'s, ..., n kinds of 1's, see example. [Joerg Arndt, May 17 2013]

			Let [x^n] F(x) denote the coefficient of x^n in F(x); then
a(0) = 1;
a(1) = [x] 1/(1-x) = 1;
a(2) = [x^2] 1/((1-x)^2*(1-x^2)) = 4;
a(3) = [x^3] 1/((1-x)^3*(1-x^2)^2*(1-x^3)) = 17;
a(4) = [x^4] 1/((1-x)^4*(1-x^2)^3*(1-x^3)^2*(1-x^4)) = 80; ...
as illustrated below.
The coefficients in Product_{k=1..n} 1/(1-x^k)^(n-k+1) for n=0..9 begin:
n=0: [(1), 0, 0, 0, 0, 0, 0, ...];
n=1: [1,(1), 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...];
n=2: [1, 2,(4), 6, 9, 12, 16, 20, 25, 30, 36, 42, ...]; (A002620)
n=3: [1, 3, 8, (17), 33, 58, 97, 153, 233, 342, 489, 681, ...]; (A002625)
n=4: [1, 4, 13, 34, (80), 170, 339, 636, 1141, 1964, 3270, ...];
n=5: [1, 5, 19, 58, 157,(384), 874, 1869, 3803, 7408, 13907, ...];
n=6: [1, 6, 26, 90, 273, 746, (1887), 4474, 10062, 21620, ...];
n=7: [1, 7, 34, 131, 438, 1314, 3632, (9385), 22940, 53466, ...];
n=8: [1, 8, 43, 182, 663, 2158, 6445, 17944, (47139), 117842, ...];
n=9: [1, 9, 53, 244, 960, 3361, 10757, 32008, 89651, (238488), ...]; ...
where the coefficients in parenthesis start this sequence.
Incidentally, the antidiagonal sums in the above table form A206119.
From _Joerg Arndt_, May 17 2013: (Start)
There are a(3)=17 partitions of 3 into 1 kind of 3's, 2 kinds of 2's, and 3 kinds of 1's:
01:  [ 1:0  1:0  1:0  ]
02:  [ 1:0  1:0  1:1  ]
03:  [ 1:0  1:0  1:2  ]
04:  [ 1:0  1:1  1:1  ]
05:  [ 1:0  1:1  1:2  ]
06:  [ 1:0  1:2  1:2  ]
07:  [ 1:0  2:0  ]
08:  [ 1:0  2:1  ]
09:  [ 1:1  1:1  1:1  ]
10:  [ 1:1  1:1  1:2  ]
11:  [ 1:1  1:2  1:2  ]
12:  [ 1:1  2:0  ]
13:  [ 1:1  2:1  ]
14:  [ 1:2  1:2  1:2  ]
15:  [ 1:2  2:0  ]
16:  [ 1:2  2:1  ]
17:  [ 3:0  ]
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A206119, A206229.

Table[SeriesCoefficient[Product[1/(1 - x^k)^(n-k+1), {k, 1, n}], {x, 0, n}], {n, 0, 40}] (* Vaclav Kotesovec, Aug 21 2018 *)

{a(n)=polcoeff(prod(k=1,n,1/(1-x^k+x*O(x^n))^(n-k+1)),n)}
for(n=0,41,print1(a(n),", "))

a(n) ~ c * d^n / sqrt(n), where d = A270915 = 5.3527013334866426877724158141653278798514832712869470973196907560641... and c = 0.2030089852709942695768237484498370155967795685257713505678384193773498... - Vaclav Kotesovec, Aug 21 2018

A259325 Infinite square array T(n,k) read by antidiagonals, defined by T(n,k) = T(n,k-1)+T(n-k,k), T(0,k)=1 (n >= 0, k >= 1).

Original entry on oeis.org

1, 1, 3, 1, 3, 7, 1, 3, 8, 13, 1, 3, 8, 16, 22, 1, 3, 8, 17, 30, 34, 1, 3, 8, 17, 33, 50, 50, 1, 3, 8, 17, 34, 58, 80, 70, 1, 3, 8, 17, 34, 61, 97, 120, 95

Offset: 0

N. J. A. Sloane, Jun 24 2015

			The first few antidiagonals are:
1
1,3,
1,3,7
1,3,8,13
1,3,8,16,22
1,3,8,17,30,34
1,3,8,17,33,50,50
1,3,8,17,34,58,80,70
1,3,8,17,34,61,97,120,95
...

E. Fix and J. L. Hodges, Jr., Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312. See Table II.
E. Fix and J. L. Hodges, Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312. [Annotated scanned copy]

Columns give A002623, A002624, A002625, A002626.

cp's OEIS Frontend

A097701 Expansion of 1/((1-x)^2*(1-x^2)^2*(1-x^3)).

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A164680 Expansion of x/((1-x)^3*(1-x^2)^3*(1-x^3)).

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A206228 a(n) = [x^n] Product_{k=1..n} 1/(1 - x^k)^(n-k+1).

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A259325 Infinite square array T(n,k) read by antidiagonals, defined by T(n,k) = T(n,k-1)+T(n-k,k), T(0,k)=1 (n >= 0, k >= 1).

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A097701 Expansion of 1/((1-x)^2(1-x^2)^2(1-x^3)).

A164680 Expansion of x/((1-x)^3(1-x^2)^3(1-x^3)).