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

Author

N. J. A. Sloane, Dec 11 1999

Keywords

• nonn
• easy

Comments

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

Examples

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

References

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

Links

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

Crossrefs

Cf. A001304, A008669.

Programs

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 *)

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 */

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