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Column 5 of array A195825. Also column 1 of triangle A195839. Also 1 together with the row sums of triangle A195839.
17

%I #47 May 17 2024 15:24:55

%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 (GW-BASIC)' A program with two A-numbers by _Omar E. Pol_, Jun 10 2012

%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

%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