A029347 Expansion of 1/((1-x^4)*(1-x^6)*(1-x^7)*(1-x^8)).

1, 0, 0, 0, 1, 0, 1, 1, 2, 0, 1, 1, 3, 1, 3, 2, 4, 1, 4, 3, 6, 3, 6, 4, 8, 4, 8, 6, 11, 6, 11, 8, 14, 8, 14, 11, 18, 11, 18, 14, 22, 14, 23, 18, 27, 18, 28, 22, 33, 23, 34, 27, 39, 28, 41, 33, 47, 34, 48, 39, 55, 41, 57, 47, 64, 48, 66, 55, 74, 57, 77, 64, 85, 66, 88, 74, 97, 77, 101, 85, 110, 88, 114

Offset

0,9

Comments

Number of partitions of n into parts 4, 6, 7, and 8. - Hoang Xuan Thanh, May 08 2026

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

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

Formula

a(n) = floor((n^3+48*n^2+564*n+3456)/8064 - (n mod 2)*(n^2+25*n+176)/384 + ((n^2+n+2) mod 4)*n/64 + ((5*n^3+2*n^2+6*n+4) mod 7)/7). - Hoang Xuan Thanh, May 08 2026

Mathematica

CoefficientList[Series[1/((1-x^4)(1-x^6)(1-x^7)(1-x^8)), {x, 0, 100}], x] (* Jinyuan Wang, Mar 11 2020 *)
(* Alternative: *)
LinearRecurrence[{0, 0, 0, 1, 0, 1, 1, 1, 0, -1, -1, -1, -1, -1, -1, 0, 1, 1, 1, 0, 1, 0, 0, 0, -1}, {1, 0, 0, 0, 1, 0, 1, 1, 2, 0, 1, 1, 3, 1, 3, 2, 4, 1, 4, 3, 6, 3, 6, 4, 8}, 70] (* Harvey P. Dale, Oct 16 2025 *)
(* Alternative: *)
a[n_]:=Floor[(n^3+48*n^2+564*n+3456)/8064-Mod[n, 2]*(n^2+25*n+176) /384+Mod[n^2+n+2, 4]*n/64+Mod[5*n^3+2*n^2+6*n+4, 7]/7]; Table[a[n], {n, 0, 100}] (* Vincenzo Librandi, May 09 2026 *)

Prog

(Magma) m:=90; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1/((1-x^4)*(1-x^6)*(1-x^7)*(1-x^8))))); // Vincenzo Librandi, May 09 2026

Crossrefs

Sequence in context: A213910 A288002 A140129 * A303427 A176076 A058725

Adjacent sequences: A029344 A029345 A029346 * A029348 A029349 A029350

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Vincenzo Librandi, May 09 2026

Status

approved