A319206 a(n) = 1*2*3*4*5 + 6*7*8*9*10 + 11*12*13*14*15 + ... + (up to n).

1, 2, 6, 24, 120, 126, 162, 456, 3144, 30360, 30371, 30492, 32076, 54384, 390720, 390736, 390992, 395616, 483744, 2251200, 2251221, 2251662, 2261826, 2506224, 8626800, 8626826, 8627502, 8646456, 9196824, 25727520, 25727551, 25728512, 25760256, 26840544

Offset: 1

Author

Wesley Ivan Hurt, Sep 13 2018

Keywords

• nonn
• easy

Comments

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.

Examples

			a(1) = 1;
a(2) = 1*2 = 2;
a(3) = 1*2*3 = 6;
a(4) = 1*2*3*4 = 24;
a(5) = 1*2*3*4*5 = 120;
a(6) = 1*2*3*4*5 + 6 = 126;
a(7) = 1*2*3*4*5 + 6*7 = 162;
a(8) = 1*2*3*4*5 + 6*7*8 = 456;
a(9) = 1*2*3*4*5 + 6*7*8*9 = 3144;
a(10) = 1*2*3*4*5 + 6*7*8*9*10 = 30360;
a(11) = 1*2*3*4*5 + 6*7*8*9*10 + 11 = 30371;
a(12) = 1*2*3*4*5 + 6*7*8*9*10 + 11*12 = 30492; etc.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, 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).

Crossrefs

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.

Programs

Mathematica

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 *)

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

Formula

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).

From Colin Barker, Sep 14 2018: (Start)

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).

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.

(End)