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A319873 a(n) = 9*8*7*6*5*4*3*2*1 + 18*17*16*15*14*13*12*11*10 + ... + (up to the n-th term).

%I A319873 #19 Oct 18 2018 10:54:40
%S A319873 9,72,504,3024,15120,60480,181440,362880,362880,362898,363186,367776,
%T A319873 436320,1391040,13728960,160755840,1764685440,17643588480,17643588507,
%U A319873 17643589182,17643606030,17644009680,17653276080,17856715680,22119259680,107157012480
%N A319873 a(n) = 9*8*7*6*5*4*3*2*1 + 18*17*16*15*14*13*12*11*10 + ... + (up to the n-th term).
%C A319873 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=9.
%H A319873 Colin Barker, <a href="/A319873/b319873.txt">Table of n, a(n) for n = 1..1000</a>
%e A319873 a(1) = 9;
%e A319873 a(2) = 9*8 = 72;
%e A319873 a(3) = 9*8*7 = 504;
%e A319873 a(4) = 9*8*7*6 = 3024;
%e A319873 a(5) = 9*8*7*6*5 = 15120;
%e A319873 a(6) = 9*8*7*6*5*4 = 60480;
%e A319873 a(7) = 9*8*7*6*5*4*3 = 181440;
%e A319873 a(8) = 9*8*7*6*5*4*3*2 = 362880;
%e A319873 a(9) = 9*8*7*6*5*4*3*2*1 = 362880;
%e A319873 a(10) = 9*8*7*6*5*4*3*2*1 + 18 = 362898;
%e A319873 a(11) = 9*8*7*6*5*4*3*2*1 + 18*17 = 363186;
%e A319873 a(12) = 9*8*7*6*5*4*3*2*1 + 18*17*16 = 367776
%e A319873 a(13) = 9*8*7*6*5*4*3*2*1 + 18*17*16*15 = 436320;
%e A319873 a(14) = 9*8*7*6*5*4*3*2*1 + 18*17*16*15*14 = 1391040;
%e A319873 a(15) = 9*8*7*6*5*4*3*2*1 + 18*17*16*15*14*13 = 13728960;
%e A319873 a(16) = 9*8*7*6*5*4*3*2*1 + 18*17*16*15*14*13*12 = 160755840;
%e A319873 a(17) = 9*8*7*6*5*4*3*2*1 + 18*17*16*15*14*13*12*11 = 1764685440;
%e A319873 a(18) = 9*8*7*6*5*4*3*2*1 + 18*17*16*15*14*13*12*11*10 = 17643588480;
%e A319873 a(19) = 9*8*7*6*5*4*3*2*1 + 18*17*16*15*14*13*12*11*10 + 27 = 17643588507;
%e A319873 etc.
%p A319873 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,9),n=1..30); # _Muniru A Asiru_, Sep 30 2018
%t A319873 k:=9; 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 A319873 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), A319872 (k=8), this sequence (k=9), A319874 (k=10).
%K A319873 nonn,easy
%O A319873 1,1
%A A319873 _Wesley Ivan Hurt_, Sep 30 2018