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%I #20 May 04 2021 15:28:02
%S 5,20,60,120,120,130,210,840,5160,30360,30375,30570,33090,63120,
%T 390720,390740,391100,397560,507000,2251200,2251225,2251800,2265000,
%U 2554800,8626800,8626830,8627670,8651160,9284520,25727520,25727555,25728710,25766790,26984160
%N a(n) = 5*4*3*2*1 + 10*9*8*7*6 + 15*14*13*12*11 + ... + (up to the n-th term).
%C 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=5.
%H Colin Barker, <a href="/A319869/b319869.txt">Table of n, a(n) for n = 1..1000</a>
%e a(1) = 5;
%e a(2) = 5*4 = 20;
%e a(3) = 5*4*3 = 60;
%e a(4) = 5*4*3*2 = 120;
%e a(5) = 5*4*3*2*1 = 120;
%e a(6) = 5*4*3*2*1 + 10 = 130;
%e a(7) = 5*4*3*2*1 + 10*9 = 210;
%e a(8) = 5*4*3*2*1 + 10*9*8 = 840;
%e a(9) = 5*4*3*2*1 + 10*9*8*7 = 5160;
%e a(10) = 5*4*3*2*1 + 10*9*8*7*6 = 30360;
%e a(11) = 5*4*3*2*1 + 10*9*8*7*6 + 15 = 30375;
%e a(12) = 5*4*3*2*1 + 10*9*8*7*6 + 15*14 = 30570;
%e a(13) = 5*4*3*2*1 + 10*9*8*7*6 + 15*14*13 = 33090;
%e a(14) = 5*4*3*2*1 + 10*9*8*7*6 + 15*14*13*12 = 63120;
%e a(15) = 5*4*3*2*1 + 10*9*8*7*6 + 15*14*13*12*11 = 390720;
%e a(16) = 5*4*3*2*1 + 10*9*8*7*6 + 15*14*13*12*11 + 20 = 390740;
%e a(17) = 5*4*3*2*1 + 10*9*8*7*6 + 15*14*13*12*11 + 20*19 = 391100;
%e a(18) = 5*4*3*2*1 + 10*9*8*7*6 + 15*14*13*12*11 + 20*19*18 = 397560;
%e a(19) = 5*4*3*2*1 + 10*9*8*7*6 + 15*14*13*12*11 + 20*19*18*17 = 507000;
%e a(20) = 5*4*3*2*1 + 10*9*8*7*6 + 15*14*13*12*11 + 20*19*18*17*16 = 2251200;
%e etc.
%p 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,5),n=1..40); # _Muniru A Asiru_, Sep 30 2018
%t k:=5; 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 For similar sequences, see: A000217 (k=1), A319866 (k=2), A319867 (k=3), A319868 (k=4), this sequence (k=5), A319870 (k=6), A319871 (k=7), A319872 (k=8), A319873 (k=9), A319874 (k=10).
%K nonn,easy
%O 1,1
%A _Wesley Ivan Hurt_, Sep 29 2018
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