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A133722 Column 3 of triangle in A133721.

%I A133722 #14 Apr 09 2020 14:37:19
%S A133722 0,0,1,1,1,1,7,3,1,25,6,1,65,10,1,140,15,1,266,21,1,462,28,1,750,36,1,
%T A133722 1155,45,1,1705,55,1,2431,66,1,3367,78,1,4550,91,1,6020,105,1,7820,
%U A133722 120,1,9996,136,1,12597,153,1,15675
%N A133722 Column 3 of triangle in A133721.
%F A133722 Conjectures from _Colin Barker_, Apr 03 2020: (Start)
%F A133722 G.f.: x^3*(1 + x + x^2 - 4*x^3 + 2*x^4 - 2*x^5 + 6*x^6 + x^8 - 4*x^9 + x^12) / ((1 - x)^5*(1 + x + x^2)^5).
%F A133722 a(n) = 5*a(n-3) - 10*a(n-6) + 10*a(n-9) - 5*a(n-12) + a(n-15) for n>14.
%F A133722 (End)
%p A133722 A133722 := proc(n)
%p A133722         A133721(n,3) ;
%p A133722 end proc:
%p A133722 seq(A133722(n),n=1..60) ; # _R. J. Mathar_, Nov 23 2011
%t A133722 A133713[l_, cl_] := Module[{g, k, s}, g = 1; For[k = 1, k <= cl + 1, k++, s = Sum[Binomial[Binomial[l, k + 1] + i - 1, i]*t^(i*k), {i, 0, Ceiling[ cl/k]}]; g = g*s]; SeriesCoefficient[g, {t, 0, cl}]];
%t A133722 a[m_, n_] := A133713[Ceiling[m/n], n*Ceiling[m/n] - m];
%t A133722 Table[a[m, 3], {m, 1, 55}] (* _Jean-François Alcover_, Apr 03 2020, after _R. J. Mathar_ *)
%K A133722 nonn
%O A133722 1,7
%A A133722 _N. J. A. Sloane_, Dec 30 2007