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A195849 Column 5 of array A195825. Also column 1 of triangle A195839. Also 1 together with the row sums of triangle A195839.

%I A195849 #51 Aug 13 2025 23:58:01
%S A195849 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 A195849 54,67,77,84,88,95,107,128,152,174,188,200,215,242,281,329,370,402,
%U A195849 428,462,513,589,674,754,816,873,940,1041,1176,1333,1477,1600,1710,1845
%N A195849 Column 5 of array A195825. Also column 1 of triangle A195839. Also 1 together with the row sums of triangle A195839.
%C A195849 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 A195849 Number of partitions of n into parts congruent to 0, 1 or 6 (mod 7). - _Ludovic Schwob_, Aug 05 2021
%H A195849 Ludovic Schwob, <a href="/A195849/b195849.txt">Table of n, a(n) for n = 0..10000</a>
%F A195849 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 A195849 a(n) ~ exp(Pi*sqrt(2*n/7)) / (8*sin(Pi/7)*n). - _Vaclav Kotesovec_, Aug 14 2017
%p A195849 A118277 := proc(n)
%p A195849         7*n^2/8+7*n/8-3/16+3*(-1)^n*(1/16+n/8) ;
%p A195849 end proc:
%p A195849 A195839 := proc(n, k)
%p A195849         option remember;
%p A195849         local ks, a, j ;
%p A195849         if A118277(k) > n then
%p A195849                 0 ;
%p A195849         elif n <= 5 then
%p A195849                 return 1;
%p A195849         elif k = 1 then
%p A195849                 a := 0 ;
%p A195849                 for j from 1 do
%p A195849                         if A118277(j) <= n-1 then
%p A195849                                 a := a+procname(n-1, j) ;
%p A195849                         else
%p A195849                                 break;
%p A195849                         end if;
%p A195849                 end do;
%p A195849                 return a;
%p A195849         else
%p A195849                 ks := A118277(k) ;
%p A195849                 (-1)^floor((k-1)/2)*procname(n-ks+1, 1) ;
%p A195849         end if;
%p A195849 end proc:
%p A195849 A195849 := proc(n)
%p A195849         A195839(n+1,1) ;
%p A195849 end proc:
%p A195849 seq(A195849(n), n=0..60) ; # _R. J. Mathar_, Oct 08 2011
%t A195849 m = 61;
%t A195849 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 A195849 (GW-BASIC)' A program with two A-numbers by _Omar E. Pol_, Jun 10 2012
%o A195849 10 Dim A118277(100), A057077(100), a(100): a(0)=1
%o A195849 20 For n = 1 to 61: For j = 1 to n
%o A195849 30 If A118277(j) <= n then a(n) = a(n) + A057077(j-1)*a(n - A118277(j))
%o A195849 40 Next j: Print a(n-1);: Next n
%Y A195849 Cf. A000041, A001082, A006950, A036820, A057077, A118277, A195825, A195829, A195839, A195848, A195850, A195851, A195852, A196933, A210843, A210964, A211971.
%K A195849 nonn
%O A195849 0,7
%A A195849 _Omar E. Pol_, Oct 07 2011