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A319208 a(n) = 1*2*3*4*5*6*7 + 8*9*10*11*12*13*14 + 15*16*17*18*19*20*21 + ... + (up to n).

%I A319208 #24 Aug 02 2020 12:00:22
%S A319208 1,2,6,24,120,720,5040,5048,5112,5760,12960,100080,1240560,17302320,
%T A319208 17302335,17302560,17306400,17375760,18697680,45209520,603353520,
%U A319208 603353542,603354026,603365664,603657120,611247120,816480720,6570915120,6570915149,6570915990
%N A319208 a(n) = 1*2*3*4*5*6*7 + 8*9*10*11*12*13*14 + 15*16*17*18*19*20*21 + ... + (up to n).
%C A319208 In general, for sequences that multiply the first k natural numbers, and then add the product of the next k natural numbers (preserving the order of operations up to n), we have a(n) = Sum_{i=1..floor(n/k)} (k*i)!/(k*i-k)! + Sum_{j=1..k-1} (1-sign((n-j) mod k)) * (Product_{i=1..j} n-i+1). Here, k=7.
%H A319208 Colin Barker, <a href="/A319208/b319208.txt">Table of n, a(n) for n = 1..1000</a>
%F A319208 a(n) = Sum_{i=1..floor(n/7)} (7*i)!/(7*i-7)! + Sum_{j=1..6} (1-sign((n-j) mod 7)) * (Product_{i=1..j} n-i+1).
%e A319208 a(1) = 1;
%e A319208 a(2) = 1*2 = 2;
%e A319208 a(3) = 1*2*3 = 6;
%e A319208 a(4) = 1*2*3*4 = 24;
%e A319208 a(5) = 1*2*3*4*5 = 120;
%e A319208 a(6) = 1*2*3*4*5*6 = 720;
%e A319208 a(7) = 1*2*3*4*5*6*7 = 5040;
%e A319208 a(8) = 1*2*3*4*5*6*7 + 8 = 5048;
%e A319208 a(9) = 1*2*3*4*5*6*7 + 8*9 = 5112;
%e A319208 a(10) = 1*2*3*4*5*6*7 + 8*9*10 = 5760;
%e A319208 a(11) = 1*2*3*4*5*6*7 + 8*9*10*11 = 12960;
%e A319208 a(12) = 1*2*3*4*5*6*7 + 8*9*10*11*12 = 100080;
%e A319208 a(13) = 1*2*3*4*5*6*7 + 8*9*10*11*12*13 = 1240560;
%e A319208 a(14) = 1*2*3*4*5*6*7 + 8*9*10*11*12*13*14 = 17302320;
%e A319208 a(15) = 1*2*3*4*5*6*7 + 8*9*10*11*12*13*14 + 15 = 17302335;
%e A319208 a(16) = 1*2*3*4*5*6*7 + 8*9*10*11*12*13*14 + 15*16 = 17302560; etc.
%t A319208 Table[Total[Times@@@Partition[Range[n],UpTo[7]]],{n,30}] (* _Harvey P. Dale_, Aug 02 2020 *)
%Y A319208 Cf. A093361, (k=1) A000217, (k=2) A228958, (k=3) A319014, (k=4) A319205, (k=5) A319206, (k=6) A319207, (k=7) this sequence, (k=8) A319209, (k=9) A319211, (k=10) A319212.
%K A319208 nonn,easy
%O A319208 1,2
%A A319208 _Wesley Ivan Hurt_, Sep 13 2018