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%I #44 Aug 06 2021 02:22:55
%S 1,1,1,1,1,1,2,3,4,4,4,4,5,7,10,12,13,13,14,16,21,27,32,34,36,38,44,
%T 54,67,77,84,88,95,107,128,152,174,188,200,215,242,281,329,370,402,
%U 428,462,513,589,674,754,816,873,940,1041,1176,1333,1477,1600,1710,1845
%N Column 5 of array A195825. Also column 1 of triangle A195839. Also 1 together with the row sums of triangle A195839.
%C Note that this sequence contains three plateaus: [1, 1, 1, 1, 1, 1], [4, 4, 4, 4], [13, 13]. For more information see A210843. See also other columns of A195825. - _Omar E. Pol_, Jun 29 2012
%C Number of partitions of n into parts congruent to 0, 1 or 6 (mod 7). - _Ludovic Schwob_, Aug 05 2021
%H Ludovic Schwob, <a href="/A195849/b195849.txt">Table of n, a(n) for n = 0..10000</a>
%F G.f.: Product_{k>=1} 1/((1 - x^(7*k))*(1 - x^(7*k-1))*(1 - x^(7*k-6))). - _Ilya Gutkovskiy_, Aug 13 2017
%F a(n) ~ exp(Pi*sqrt(2*n/7)) / (8*sin(Pi/7)*n). - _Vaclav Kotesovec_, Aug 14 2017
%p A118277 := proc(n)
%p 7*n^2/8+7*n/8-3/16+3*(-1)^n*(1/16+n/8) ;
%p end proc:
%p A195839 := proc(n, k)
%p option remember;
%p local ks, a, j ;
%p if A118277(k) > n then
%p 0 ;
%p elif n <= 5 then
%p return 1;
%p elif k = 1 then
%p a := 0 ;
%p for j from 1 do
%p if A118277(j) <= n-1 then
%p a := a+procname(n-1, j) ;
%p else
%p break;
%p end if;
%p end do;
%p return a;
%p else
%p ks := A118277(k) ;
%p (-1)^floor((k-1)/2)*procname(n-ks+1, 1) ;
%p end if;
%p end proc:
%p A195849 := proc(n)
%p A195839(n+1,1) ;
%p end proc:
%p seq(A195849(n), n=0..60) ; # _R. J. Mathar_, Oct 08 2011
%t m = 61;
%t Product[1/((1 - x^(7k))(1 - x^(7k - 1))(1 - x^(7k - 6))), {k, 1, m}] + O[x]^m // CoefficientList[#, x]& ( _Jean-François Alcover_, Apr 13 2020, after _Ilya Gutkovskiy_ *)
%o From _Omar E. Pol_, Jun 10 2012: (Start)
%o (GWbasic)' A program with two A-numbers:
%o 10 Dim A118277(100), A057077(100), a(100): a(0)=1
%o 20 For n = 1 to 61: For j = 1 to n
%o 30 If A118277(j) <= n then a(n) = a(n) + A057077(j-1)*a(n - A118277(j))
%o 40 Next j: Print a(n-1);: Next n (End)
%Y Cf. A000041, A001082, A006950, A036820, A057077, A118277, A195825, A195829, A195839, A195848, A195850, A195851, A195852, A196933, A210843, A210964, A211971.
%K nonn
%O 0,7
%A _Omar E. Pol_, Oct 07 2011
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