A319207 a(n) = 1*2*3*4*5*6 + 7*8*9*10*11*12 + 13*14*15*16*17*18 + ... + (up to n).

1, 2, 6, 24, 120, 720, 727, 776, 1224, 5760, 56160, 666000, 666013, 666182, 668730, 709680, 1408560, 14032080, 14032099, 14032460, 14040060, 14207640, 18069960, 110941200, 110941225, 110941850, 110958750, 111432600, 125191800, 538459200, 538459231, 538460192

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=6.

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 = 720;
a(7) = 1*2*3*4*5*6 + 7 = 727;
a(8) = 1*2*3*4*5*6 + 7*8 = 776;
a(9) = 1*2*3*4*5*6 + 7*8*9 = 1224;
a(10) = 1*2*3*4*5*6 + 7*8*9*10 = 5760;
a(11) = 1*2*3*4*5*6 + 7*8*9*10*11 = 56160;
a(12) = 1*2*3*4*5*6 + 7*8*9*10*11*12 = 666000;
a(13) = 1*2*3*4*5*6 + 7*8*9*10*11*12 + 13 = 666013;
a(14) = 1*2*3*4*5*6 + 7*8*9*10*11*12 + 13*14 = 666182; etc.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,7,-7,0,0,0,0,-21,21,0,0,0,0,35,-35,0,0,0,0,-35,35,0,0,0,0,21,-21,0,0,0,0,-7,7,0,0,0,0,1,-1).

Crossrefs

Cf. A093361, (k=1) A000217, (k=2) A228958, (k=3) A319014, (k=4) A319205, (k=5) A319206, (k=6) this sequence, (k=7) A319208, (k=8) A319209, (k=9) A319211, (k=10) A319212.

Programs

Mathematica

a[n_]:=Sum[(6*i)!/(6*i-6)!, {i, 1, Floor[n/6] }] + Sum[(1-Sign[Mod[n-j,6]])*Product[n-i+1, {i, 1, j}], {j, 1, 5}] ; Array[a, 40] (* Stefano Spezia, Sep 17 2018 *)

PARI

Vec(x*(1 + x + 4*x^2 + 18*x^3 + 96*x^4 + 600*x^5 + 42*x^7 + 420*x^8 + 4410*x^9 + 49728*x^10 + 605640*x^11 - 15*x^12 - 153*x^13 - 504*x^14 + 9576*x^15 + 348096*x^16 + 8367240*x^17 + 40*x^18 + 172*x^19 - 968*x^20 - 24444*x^21 + 25200*x^22 + 17292240*x^23 - 45*x^24 - 33*x^25 + 1668*x^26 + 2610*x^27 - 361200*x^28 + 6939240*x^29 + 24*x^30 - 54*x^31 - 540*x^32 + 7650*x^33 - 61680*x^34 + 387240*x^35 - 5*x^36 + 25*x^37 - 80*x^38 + 180*x^39 - 240*x^40 + 120*x^41) / ((1 - x)^8*(1 + x)^7*(1 - x + x^2)^7*(1 + x + x^2)^7) + O(x^40)) \\ Colin Barker, Sep 14 2018

Formula

a(n) = Sum_{i=1..floor(n/6)} (6*i)!/(6*i-6)! + Sum_{j=1..5} (1-sign((n-j) mod 6)) * (Product_{i=1..j} n-i+1).

G.f.: x*(1 + x + 4*x^2 + 18*x^3 + 96*x^4 + 600*x^5 + 42*x^7 + 420*x^8 + 4410*x^9 + 49728*x^10 + 605640*x^11 - 15*x^12 - 153*x^13 - 504*x^14 + 9576*x^15 + 348096*x^16 + 8367240*x^17 + 40*x^18 + 172*x^19 - 968*x^20 - 24444*x^21 + 25200*x^22 + 17292240*x^23 - 45*x^24 - 33*x^25 + 1668*x^26 + 2610*x^27 - 361200*x^28 + 6939240*x^29 + 24*x^30 - 54*x^31 - 540*x^32 + 7650*x^33 - 61680*x^34 + 387240*x^35 - 5*x^36 + 25*x^37 - 80*x^38 + 180*x^39 - 240*x^40 + 120*x^41) / ((1 - x)^8*(1 + x)^7*(1 - x + x^2)^7*(1 + x + x^2)^7). - Colin Barker, Sep 14 2018

a(n) = a(n-1) + 7*a(n-6) - 7*a(n-7) - 21*a(n-12) + 21*a(n-13) + 35*a(n-18) - 35*a(n-19) - 35*a(n-24) + 35*a(n-25) + 21*a(n-30) - 21*a(n-31) - 7*a(n-36) + 7*a(n-37) + a(n-42) - a(n-43). - Wesley Ivan Hurt, Jun 20 2024