A025876 Expansion of 1/((1-x^5)*(1-x^6)*(1-x^7)).

1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 7, 6, 6, 7, 7, 8, 8, 8, 8, 8, 9, 9, 10, 10, 10, 10, 10, 11, 12, 12, 12, 12, 12, 13, 14, 14, 14, 14, 15, 15, 16, 16, 16, 17, 17, 18, 18

Offset

0,13

Comments

With a(0)=0, a(n) is the number of partitions of n into 4 parts whose largest part is twice the smallest part. - Wesley Ivan Hurt, Jan 06 2021

a(n) is the number of partitions of n into parts 5, 6, and 7. - Joerg Arndt, Jan 06 2021

Links

G. C. Greubel, Table of n, a(n) for n = 0..5000

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1,1,1,0,0,0,-1,-1,-1,0,0,0,0,1).

Formula

a(n) = a(n-5) +a(n-6) +a(n-7) -a(n-11) -a(n-12) -a(n-13) +a(n-18). - Harvey P. Dale, Dec 16 2013

For n > 0, a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} [3*k = n-i-j], where [ ] is the Iverson bracket. - Wesley Ivan Hurt, Jan 06 2021

a(n) = floor((n^2 + 18*n + 165)/420 + ((n mod 6) - 3)^2/12). - Hoang Xuan Thanh, Sep 14 2025

Mathematica

CoefficientList[Series[1/((1-x^5)(1-x^6)(1-x^7)), {x, 0, 80}], x] (* or *)
LinearRecurrence[{0, 0, 0, 0, 1, 1, 1, 0, 0, 0, -1, -1, -1, 0, 0, 0, 0, 1}, {1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 2, 1, 1, 1, 1, 2}, 80] (* Harvey P. Dale, Dec 16 2013 *)

Prog

(Magma)
R<x>:=PowerSeriesRing(Rationals(), 80);
Coefficients(R!( 1/((1-x^5)*(1-x^6)*(1-x^7)) )); // G. C. Greubel, Nov 17 2022
(SageMath)
def A025876_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x^5)*(1-x^6)*(1-x^7)) ).list()
A025876_list(80) # G. C. Greubel, Nov 17 2022
(PARI) Vec(1/((1-x^5)*(1-x^6)*(1-x^7))+O(x^79)) \\ Stefano Spezia, Sep 17 2025

Crossrefs

Cf. A025877, A025878, A025879, A025880, A025881.

Sequence in context: A328523 A025887 A025882 * A109035 A244231 A237706

Adjacent sequences: A025873 A025874 A025875 * A025877 A025878 A025879

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved