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A069982
Number of 4-gonal compositions of n into positive parts.
2
0, 0, 0, 0, 1, 4, 10, 16, 31, 40, 68, 80, 125, 140, 206, 224, 315, 336, 456, 480, 633, 660, 850, 880, 1111, 1144, 1420, 1456, 1781, 1820, 2198, 2240, 2675, 2720, 3216, 3264, 3825, 3876, 4506, 4560, 5263, 5320, 6100, 6160, 7021, 7084, 8030
OFFSET
0,6
LINKS
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, European Journal of Combinatorics, Vol. 22, No. 7 (2001), 887-904.
FORMULA
G.f.: q^4/(1-q)^4-4*q^7/(1-q)^4/(1+q)^3.
a(n) = (2*n^3-3*n^2-23*n+3*(13+(n^2-7*n+11)*(-1)^n))/24. - Luce ETIENNE, Jul 02 2015; edited by Mo Li, Sep 18 2019
a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7) for n>7. - Colin Barker, Sep 18 2019
MATHEMATICA
Table[Piecewise[{
{Binomial[k - 1, k - 4] - 4*Binomial[(k - 1)/2, (k - 7)/2], Mod[k, 2] == 1},
{Binomial[k - 1, k - 4] - 4*Binomial[(k - 2)/2, (k - 8)/2], Mod[k, 2] == 0}}], {k, 1, 20}] (* Mo Li, Sep 18 2019 *)
PROG
(PARI) concat([0, 0, 0, 0], Vec(x^4*(1 + 3*x + 3*x^2 - 3*x^3) / ((1 - x)^4*(1 + x)^3) + O(x^40))) \\ Colin Barker, Sep 18 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, May 06 2002
STATUS
approved