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A319872 a(n) = 8*7*6*5*4*3*2*1 + 16*15*14*12*11*10*9 + ... + (up to the n-th term).

%I A319872 #19 Oct 18 2018 10:54:48
%S A319872 8,56,336,1680,6720,20160,40320,40320,40336,40560,43680,84000,564480,
%T A319872 5806080,57697920,518958720,518958744,518959272,518970864,519213744,
%U A319872 524059200,615867840,2263322880,30173149440,30173149472,30173150432,30173179200,30174012480
%N A319872 a(n) = 8*7*6*5*4*3*2*1 + 16*15*14*12*11*10*9 + ... + (up to the n-th term).
%C A319872 For similar multiply/add sequences in descending blocks of k natural numbers, we have: a(n) = Sum_{j=1..k-1} (floor((n-j)/k)-floor((n-j-1)/k)) * (Product_{i=1..j} n-i-j+k+1) + Sum_{j=1..n} (floor(j/k)-floor((j-1)/k)) * (Product_{i=1..k} j-i+1). Here, k=8.
%H A319872 Colin Barker, <a href="/A319872/b319872.txt">Table of n, a(n) for n = 1..1000</a>
%e A319872 a(1) = 8;
%e A319872 a(2) = 8*7 = 56;
%e A319872 a(3) = 8*7*6 = 336;
%e A319872 a(4) = 8*7*6*5 = 1680;
%e A319872 a(5) = 8*7*6*5*4 = 6720;
%e A319872 a(6) = 8*7*6*5*4*3 = 20160;
%e A319872 a(7) = 8*7*6*5*4*3*2 = 40320;
%e A319872 a(8) = 8*7*6*5*4*3*2*1 = 40320;
%e A319872 a(9) = 8*7*6*5*4*3*2*1 + 16 = 40336;
%e A319872 a(10) = 8*7*6*5*4*3*2*1 + 16*15 = 40560;
%e A319872 a(11) = 8*7*6*5*4*3*2*1 + 16*15*14 = 43680;
%e A319872 a(12) = 8*7*6*5*4*3*2*1 + 16*15*14*13 = 84000;
%e A319872 a(13) = 8*7*6*5*4*3*2*1 + 16*15*14*13*12 = 564480;
%e A319872 a(14) = 8*7*6*5*4*3*2*1 + 16*15*14*13*12*11 = 5806080;
%e A319872 a(15) = 8*7*6*5*4*3*2*1 + 16*15*14*13*12*11*10 = 57697920;
%e A319872 a(16) = 8*7*6*5*4*3*2*1 + 16*15*14*13*12*11*10*9 = 518958720;
%e A319872 a(17) = 8*7*6*5*4*3*2*1 + 16*15*14*13*12*11*10*9 + 24 = 518958744;
%e A319872 a(18) = 8*7*6*5*4*3*2*1 + 16*15*14*13*12*11*10*9 + 24*23 = 518959272;
%e A319872 a(19) = 8*7*6*5*4*3*2*1 + 16*15*14*13*12*11*10*9 + 24*23*22 = 518970864;
%e A319872 a(20) = 8*7*6*5*4*3*2*1 + 16*15*14*13*12*11*10*9 + 24*23*22*21 = 519213744;
%e A319872 etc.
%p A319872 a:=(n,k)->add((floor((n-j)/k)-floor((n-j-1)/k))*(mul(n-i-j+k+1,i=1..j)),j=1..k-1) + add((floor(j/k)-floor((j-1)/k))*(mul(j-i+1,i=1..k)),j=1..n): seq(a(n,8),n=1..30); # _Muniru A Asiru_, Sep 30 2018
%t A319872 k:=8; a[n_]:=Sum[(Floor[(n-j)/k]-Floor[(n-j-1)/k])* Product[n-i-j+k+1, {i,1,j }], {j,1,k-1} ]  + Sum[(Floor[j/k]-Floor[(j-1)/k])* Product[j-i+1, {i,1,k} ], {j,1,n}]; Array[a, 50] (* _Stefano Spezia_, Sep 30 2018 *)
%Y A319872 For similar sequences, see: A000217 (k=1), A319866 (k=2), A319867 (k=3), A319868 (k=4), A319869 (k=5), A319870 (k=6), A319871 (k=7), this sequence (k=8), A319873 (k=9), A319874 (k=10).
%K A319872 nonn,easy
%O A319872 1,1
%A A319872 _Wesley Ivan Hurt_, Sep 30 2018