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Number of 5-gonal compositions of n into positive parts.
2

%I #18 Sep 26 2019 09:41:51

%S 0,0,0,0,0,1,5,15,35,65,121,185,305,420,640,826,1190,1470,2030,2430,

%T 3246,3795,4935,5665,7205,8151,10175,11375,13975,15470,18746,20580,

%U 24640,26860,31820,34476,40460,43605,50745,54435,62871,67165

%N Number of 5-gonal compositions of n into positive parts.

%H Colin Barker, <a href="/A069983/b069983.txt">Table of n, a(n) for n = 0..1000</a>

%H G. E. Andrews, P. Paule and A. Riese, <a href="https://www.researchgate.net/publication/277299165_MacMahon&#39;s_Partition_Analysis_III_The_Omega_Package">MacMahon's partition analysis III. The Omega package</a>, p. 17.

%H G. E. Andrews, P. Paule and A. Riese, <a href="https://doi.org/10.1006/eujc.2001.0527">MacMahon's Partition Analysis: The Omega Package</a>, Europ. J. Combin., 22 (2001), 887-904.

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,-4,-6,6,4,-4,-1,1).

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

%F 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

%t Table[Piecewise[{

%t {Binomial[k - 1, k - 5] - 5*Binomial[(k - 1)/2, (k - 9)/2], Mod[k, 2] == 1},

%t {Binomial[k - 1, k - 5] - 5*Binomial[(k - 2)/2, (k - 10)/2],Mod[k, 2] == 0}

%t }], {k, 1, 20}] (* _Mo Li_, Sep 18 2019 *)

%o (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

%Y Cf. A069981, A069982, A005044.

%K nonn,easy

%O 0,7

%A _N. J. A. Sloane_, May 06 2002