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%I #20 Aug 05 2021 16:25:25
%S 1,0,0,1,1,0,1,2,1,1,3,3,2,3,6,4,4,8,9,6,10,15,12,12,21,22,18,25,36,
%T 30,32,48,52,45,60,78,72,75,105,113,105,130,166,156,166,218,236,224,
%U 274,332,325,345,436,469,462,544,649,644,688,839,907,903,1051
%N Number of partitions of n into parts congruent to 0, 3 or 4 (mod 7).
%H Ludovic Schwob, <a href="/A346798/b346798.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-3))*(1 - x^(7*k-4))).
%F a(n) = a(n-3) + a(n-4) - a(n-13) - a(n-15) + + - - (with a(0)=1 and a(n) = 0 for negative n), where 3, 4, 13, 15, ... is the sequence A057570.
%F a(n) ~ exp(Pi*sqrt(2*n/7)) / (8*cos(Pi/14)*n). - _Vaclav Kotesovec_, Aug 05 2021
%e For n=19 the a(19)=6 solutions are 3+3+3+3+3+4, 3+3+3+3+7, 3+3+3+10, 3+4+4+4+4, 4+4+4+7, and 4+4+11.
%t CoefficientList[Series[Product[1/((1 - x^(7*k))(1 - x^(7*k-3))(1 - x^(7*k-4))),{k,55}],{x,0,55}],x] (* _Stefano Spezia_, Aug 04 2021 *)
%Y Cf. A000041, A047342, A057570, A006950, A036820, A036822, A195848, A035363, A195849, A346797.
%K nonn
%O 0,8
%A _Ludovic Schwob_, Aug 04 2021
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