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

Author

Ralf Stephan, Aug 24 2004

Keywords

• nonn
• easy

Comments

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

Examples

			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
		

Links

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

Crossrefs

First differences of A002625. Partial sums of A008763.

Programs

Maple

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

Mathematica

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*)

PARI

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)))

PARI

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

Formula

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