A028291 Expansion of 1/((1-x)^2(1-x^2)(1-x^3)(1-x^5)) in powers of x.

1, 2, 4, 7, 11, 17, 25, 35, 48, 64, 84, 108, 137, 171, 211, 258, 312, 374, 445, 525, 616, 718, 832, 959, 1100, 1256, 1428, 1617, 1824, 2050, 2297, 2565, 2856, 3171, 3511, 3878, 4273, 4697, 5152, 5639, 6160, 6716, 7309, 7940, 8611, 9324, 10080, 10881, 11729

Offset

0,2

Comments

Partitions of n into parts 1, 2, 3, and 5. - Joerg Arndt, Jun 05 2014

References

Susan Elle, Ore extensions of global dimension 5, Abstract 1110-17-204, Abstracts Amer. Math. Soc., 36 (No. 2, 2015), p. 822.

Links

Table of n, a(n) for n=0..48.

Index entries for linear recurrences with constant coefficients, signature (2,0,-1,-1,1,0,-1,1,1,0,-2,1).

Formula

a(n) = round((n+1)*(n^3+23*n^2+173*n+451)/720). - Tani Akinari, Jun 05 2014

a(n) - 2*a(n-1) + a(n+3) + a(n+4) - 2*a(n+6) + a(n+7) = 1 if n == 3 (mod 5) else 0. - Michael Somos, Jun 05 2014

a(n) = a(-12 - n) for all n in Z. - Michael Somos, May 14 2015

a(n) - a(n-1) = A008669(n), a(n) - a(n-3) = A001304(n) for all n in Z. - Michael Somos, May 14 2015

Euler transform of length 5 sequence [ 2, 1, 1, 0, 1]. - Michael Somos, May 14 2015

Example

G.f. = 1 + 2*x + 4*x^2 + 7*x^3 + 11*x^4 + 17*x^5 + 25*x^6 + 35*x^7 + ...

Mathematica

a[ n_] := Quotient[n (n + 12) (n^2 + 12 n + 52), 720] + 1; (* Michael Somos, Jun 05 2014 *)
a[ n_] := With[{m = If[ n < 0, -12 - n, n]}, SeriesCoefficient[ 1 / ((1 - x)^2*(1 - x^2)*(1 - x^3)*(1 - x^5)), {x, 0, m}]]; (* Michael Somos, Jun 05 2014 *)
Table[Round[(n + 1)*(n^3 + 23*n^2 + 173*n + 451)/720], {n, 0, 40}] (* Wesley Ivan Hurt, Jun 05 2014 *)
LinearRecurrence[{2, 0, -1, -1, 1, 0, -1, 1, 1, 0, -2, 1}, {1, 2, 4, 7, 11, 17, 25, 35, 48, 64, 84, 108}, 50] (* Harvey P. Dale, Sep 06 2022 *)

Prog

(PARI) Vec(1/((1-x)^2*(1-x^2)*(1-x^3)*(1-x^5))+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012
(PARI) {a(n) = n * (n+12) * (n^2 + 12*n + 52) \ 720 + 1}; /* Michael Somos, Jun 05 2014 */
(PARI) {a(n) = if( n<0, n = -12 - n); polcoeff( 1 / ((1 - x)^2 * (1 - x^2) * (1 - x^3) * (1 - x^5)) + x * O(x^n), n)}; /* Michael Somos, Jun 05 2014 */

Crossrefs

Cf. A001304, A008669.

Sequence in context: A308825 A266901 A117649 * A067997 A175491 A034379

Adjacent sequences: A028288 A028289 A028290 * A028292 A028293 A028294

Keyword

nonn,easy

Author

N. J. A. Sloane, Dec 11 1999

Status

approved