A069983 Number of 5-gonal compositions of n into positive parts.

0, 0, 0, 0, 0, 1, 5, 15, 35, 65, 121, 185, 305, 420, 640, 826, 1190, 1470, 2030, 2430, 3246, 3795, 4935, 5665, 7205, 8151, 10175, 11375, 13975, 15470, 18746, 20580, 24640, 26860, 31820, 34476, 40460, 43605, 50745, 54435, 62871, 67165

Offset

0,7

Links

Colin Barker, Table of n, a(n) for n = 0..1000

G. E. Andrews, P. Paule and A. Riese, MacMahon's partition analysis III. The Omega package, p. 17.

G. E. Andrews, P. Paule and A. Riese, MacMahon's Partition Analysis: The Omega Package, Europ. J. Combin., 22 (2001), 887-904.

Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,6,4,-4,-1,1).

Formula

G.f.: q^5/(1-q)^5 - 5*q^9/((1-q)^5*(1+q)^4).

a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9) for n > 9. - Colin Barker, Sep 18 2019

Mathematica

Table[Piecewise[{
   {Binomial[k - 1, k - 5] - 5*Binomial[(k - 1)/2, (k - 9)/2], Mod[k, 2] == 1},
   {Binomial[k - 1, k - 5] - 5*Binomial[(k - 2)/2, (k - 10)/2], Mod[k, 2] == 0}
   }], {k, 1, 20}] (* Mo Li, Sep 18 2019 *)

Prog

(PARI) concat([0, 0, 0, 0, 0], Vec(x^5*(1 + 4*x + 6*x^2 + 4*x^3 - 4*x^4) / ((1 - x)^5*(1 + x)^4) + O(x^40))) \\ Colin Barker, Sep 18 2019

Crossrefs

Cf. A069981, A069982, A005044.

Sequence in context: A233348 A061829 A063382 * A005894 A374711 A015622

Adjacent sequences: A069980 A069981 A069982 * A069984 A069985 A069986

Keyword

nonn,easy

Author

N. J. A. Sloane, May 06 2002

Status

approved