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%I #27 Apr 20 2023 18:09:20
%S 1,2,6,24,120,126,162,456,3144,30360,30371,30492,32076,54384,390720,
%T 390736,390992,395616,483744,2251200,2251221,2251662,2261826,2506224,
%U 8626800,8626826,8627502,8646456,9196824,25727520,25727551,25728512,25760256,26840544
%N a(n) = 1*2*3*4*5 + 6*7*8*9*10 + 11*12*13*14*15 + ... + (up to n).
%C 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=5.
%H Colin Barker, <a href="/A319206/b319206.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_31">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,6,-6,0,0,0,-15,15,0,0,0,20,-20,0,0,0,-15,15,0,0,0,6,-6,0,0,0,-1,1).
%F a(n) = Sum_{i=1..floor(n/5)} (5*i)!/(5*i-5)! + Sum_{j=1..4} (1-sign((n-j) mod 5)) * (Product_{i=1..j} n-i+1).
%F From _Colin Barker_, Sep 14 2018: (Start)
%F G.f.: x*(1 + x + 4*x^2 + 18*x^3 + 96*x^4 + 30*x^6 + 270*x^7 + 2580*x^8 + 26640*x^9 - 10*x^10 - 80*x^11 - 120*x^12 + 6450*x^13 + 174480*x^14 + 20*x^15 + 50*x^16 - 550*x^17 - 5760*x^18 + 155760*x^19 - 15*x^20 + 15*x^21 + 360*x^22 - 3240*x^23 + 18000*x^24 + 4*x^25 - 16*x^26 + 36*x^27 - 48*x^28 + 24*x^29) / ((1 - x)^7*(1 + x + x^2 + x^3 + x^4)^6).
%F a(n) = a(n-1) + 6*a(n-5) - 6*a(n-6) - 15*a(n-10) + 15*a(n-11) + 20*a(n-15) - 20*a(n-16) - 15*a(n-20) + 15*a(n-21) + 6*a(n-25) - 6*a(n-26) - a(n-30) + a(n-31) for n>31.
%F (End)
%e a(1) = 1;
%e a(2) = 1*2 = 2;
%e a(3) = 1*2*3 = 6;
%e a(4) = 1*2*3*4 = 24;
%e a(5) = 1*2*3*4*5 = 120;
%e a(6) = 1*2*3*4*5 + 6 = 126;
%e a(7) = 1*2*3*4*5 + 6*7 = 162;
%e a(8) = 1*2*3*4*5 + 6*7*8 = 456;
%e a(9) = 1*2*3*4*5 + 6*7*8*9 = 3144;
%e a(10) = 1*2*3*4*5 + 6*7*8*9*10 = 30360;
%e a(11) = 1*2*3*4*5 + 6*7*8*9*10 + 11 = 30371;
%e a(12) = 1*2*3*4*5 + 6*7*8*9*10 + 11*12 = 30492; etc.
%t a[n_]:=Sum[(5*i)!/(5*i-5)!, {i, 1, Floor[n/5] }] + Sum[(1-Sign[Mod[n-j, 5]])*Product[n-i+1, {i, 1, j}], {j, 1, 4}] ; Array[a, 34] (* _Stefano Spezia_, Apr 18 2023 *)
%o (PARI) Vec(x*(1 + x + 4*x^2 + 18*x^3 + 96*x^4 + 30*x^6 + 270*x^7 + 2580*x^8 + 26640*x^9 - 10*x^10 - 80*x^11 - 120*x^12 + 6450*x^13 + 174480*x^14 + 20*x^15 + 50*x^16 - 550*x^17 - 5760*x^18 + 155760*x^19 - 15*x^20 + 15*x^21 + 360*x^22 - 3240*x^23 + 18000*x^24 + 4*x^25 - 16*x^26 + 36*x^27 - 48*x^28 + 24*x^29) / ((1 - x)^7*(1 + x + x^2 + x^3 + x^4)^6) + O(x^40)) \\ _Colin Barker_, Sep 14 2018
%Y Cf. A093361, (k=1) A000217, (k=2) A228958, (k=3) A319014, (k=4) A319205, (k=5) this sequence, (k=6) A319207, (k=7) A319208, (k=8) A319209, (k=9) A319211, (k=10) A319212.
%K nonn,easy
%O 1,2
%A _Wesley Ivan Hurt_, Sep 13 2018
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