A319212 a(n) = 1*2*3*4*5*6*7*8*9*10 + 11*12*13*14*15*16*17*18*19*20 + 21*22*23*24*25*26*27*28*29*30 + ... + (up to n).

1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 3628811, 3628932, 3630516, 3652824, 3989160, 9394560, 101646720, 1767951360, 33525757440, 670446201600, 670446201621, 670446202062, 670446212226, 670446456624, 670452577200, 670611967200, 674921872800

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=10.

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 = 5040;
a(8) = 1*2*3*4*5*6*7*8 = 40320;
a(9) = 1*2*3*4*5*6*7*8*9 = 362880;
a(10) = 1*2*3*4*5*6*7*8*9*10 = 3628800;
a(11) = 1*2*3*4*5*6*7*8*9*10 + 11 = 3628811;
a(12) = 1*2*3*4*5*6*7*8*9*10 + 11*12 = 3628932;
a(13) = 1*2*3*4*5*6*7*8*9*10 + 11*12*13 = 3630516;
a(14) = 1*2*3*4*5*6*7*8*9*10 + 11*12*13*14 = 3652824;
a(15) = 1*2*3*4*5*6*7*8*9*10 + 11*12*13*14*15 = 3989160;
a(16) = 1*2*3*4*5*6*7*8*9*10 + 11*12*13*14*15*16 = 9394560;
a(17) = 1*2*3*4*5*6*7*8*9*10 + 11*12*13*14*15*16*17 = 101646720;
a(18) = 1*2*3*4*5*6*7*8*9*10 + 11*12*13*14*15*16*17*18 = 1767951360;
a(19) = 1*2*3*4*5*6*7*8*9*10 + 11*12*13*14*15*16*17*18*19 = 33525757440;
a(20) = 1*2*3*4*5*6*7*8*9*10 + 11*12*13*14*15*16*17*18*19*20 = 670446201600;
etc.
		

Links

• Georg Fischer, Table of n, a(n) for n = 1..1000 (first 739 terms from Colin Barker)

Crossrefs

Cf. A093361, (k=1) A000217, (k=2) A228958, (k=3) A319014, (k=4) A319205, (k=5) A319206, (k=6) A319207, (k=7) A319208, (k=8) A319209, (k=9) A319211, (k=10) this sequence.

Programs

Mathematica

a[n_]:=Sum[(10*i)!/(10*i-10)!, {i, 1, Floor[n/10] }] + Sum[(1-Sign[Mod[n-j,10]])*Product[n-i+1, {i, 1, j}], {j, 1, 9}] ; Array[a, 40] (* or *)
CoefficientList[Series[x (1 + x + 4 x^2 + 18 x^3 + 96 x^4 + 600 x^5 + 4320 x^6 + 35280 x^7 + 322560 x^8 + 3265920 x^9 + 110 x^11 + 1540 x^12 + 22110 x^13 + 335280 x^14 + 5398800 x^15 + 92204640 x^16 + 1665916560 x^17 + 31754257920 x^18 + 636884519040 x^19 - 45 x^20 - 835 x^21 - 7040 x^22 + 2426160 x^24 + 99963600 x^25 + 3295369440 x^26 + 102515711760 x^27 + 3159608094720 x^28 + 98387160157440 x^29 + 240 x^30 + 2600 x^31 + 6400 x^32 - 384120 x^33 - 11000880 x^34 - 92637600 x^35 + 8150963040 x^36 + 682266206160 x^37 + 38076411985920 x^38 + 1874796686864640 x^39 - 630 x^40 - 4270 x^41 + 22120 x^42 + 1067820 x^43 + 8250000 x^44 - 525742800 x^45 - 23300782560 x^46 + 150285587760 x^47 + 93849442283520 x^48 + 9232053795296640 x^49 + 1008 x^50 + 3668 x^51 - 67928 x^52 - 1130796 x^53 + 15384048 x^54 + 861484800 x^55 - 7313090400 x^56 - 1717130091600 x^57 + 1723567507200 x^58 + 14964584346835200 x^59 - 1050 x^60 - 910 x^61 + 81760 x^62 + 291240 x^63 - 27736080 x^64 - 136792800 x^65 + 29138931360 x^66 - 117003292560 x^67 - 93887882161920 x^68 + 8480246509848960 x^69 + 720 x^70 - 1240 x^71 - 49760 x^72 + 371400 x^73 + 13094640 x^74 - 362037600 x^75 - 4579273440 x^76 + 749464032240 x^77 - 39511261278720 x^78 + 1564662885730560 x^79 - 315 x^80 + 1325 x^81 + 13300 x^82 - 301230 x^83 + 553680 x^84 + 130188600 x^85 - 5213255040 x^86 + 144639915840 x^87 - 3405078673920 x^88 + 72262987695360 x^89 + 80 x^90 - 530 x^91 + 180 x^92 + 60030 x^93 - 1288800 x^94 + 20098800 x^95 - 270829440 x^96 + 3295434240 x^97 - 36561611520 x^98 + 368739423360 x^99 - 9 x^100 + 81 x^101 - 576 x^102 + 3528 x^103 - 18144 x^104 + 75600 x^105 - 241920 x^106 + 544320 x^107 - 725760 x^108 + 362880 x^109)/((1 - x)^12 (1 + x)^11 (1 + x^2 + x^4 + x^6 + x^8)^11), {x, 0, 40}], x] (* Stefano Spezia, Sep 17 2018 *)

Formula

a(n) = Sum_{i=1..floor(n/10)} (10*i)!/(10*i-10)! + Sum_{j=1..9} (1-sign((n-j) mod 10)) * (Product_{i=1..j} n-i+1).

From Stefano Spezia, Sep 17 2018: (Start)

G.f.: x*(1 + x + 4*x^2 + 18*x^3 + 96*x^4 + 600*x^5 + 4320*x^6 + 35280*x^7 + 322560*x^8 + 3265920*x^9 + 110*x^11 + 1540*x^12 + 22110*x^13 + 335280*x^14 + 5398800*x^15 + 92204640*x^16 + 1665916560*x^17 + 31754257920*x^18 + 636884519040*x^19 - 45*x^20 - 835*x^21 - 7040*x^22 + 2426160*x^24 + 99963600*x^25 + 3295369440*x^26 + 102515711760*x^27 + 3159608094720*x^28 + 98387160157440*x^29 + 240*x^30 + 2600*x^31 + 6400*x^32 - 384120*x^33 - 11000880*x^34 - 92637600*x^35 + 8150963040*x^36 + 682266206160*x^37 + 38076411985920*x^38 + 1874796686864640*x^39 - 630*x^40 - 4270*x^41 + 22120*x^42 + 1067820*x^43 + 8250000*x^44 - 525742800*x^45 - 23300782560*x^46 + 150285587760*x^47 + 93849442283520*x^48 + 9232053795296640*x^49 + 1008*x^50 + 3668*x^51 - 67928*x^52 - 1130796*x^53 + 15384048*x^54 + 861484800*x^55 - 7313090400*x^56 - 1717130091600*x^57 + 1723567507200*x^58 + 14964584346835200*x^59 - 1050*x^60 - 910*x^61 + 81760*x^62 + 291240*x^63 - 27736080*x^64 - 136792800*x^65 + 29138931360*x^66 - 117003292560*x^67 - 93887882161920*x^68 + 8480246509848960*x^69 + 720*x^70 - 1240*x^71 - 49760*x^72 + 371400*x^73 + 13094640*x^74 - 362037600*x^75 - 4579273440*x^76 + 749464032240*x^77 - 39511261278720*x^78 + 1564662885730560*x^79 - 315*x^80 + 1325*x^81 + 13300*x^82 - 301230*x^83 + 553680*x^84 + 130188600*x^85 - 5213255040*x^86 + 144639915840*x^87 - 3405078673920*x^88 + 72262987695360*x^89 + 80*x^90 - 530*x^91 + 180*x^92 + 60030*x^93 - 1288800*x^94 + 20098800*x^95 - 270829440*x^96 + 3295434240*x^97 - 36561611520*x^98 + 368739423360*x^99 - 9*x^100 + 81*x^101 - 576*x^102 + 3528*x^103 - 18144*x^104 + 75600*x^105 - 241920*x^106 + 544320*x^107 - 725760*x^108 + 362880*x^109)/((1 - x)^12*(1 + x)^11*(1 + x^2 + x^4 + x^6 + x^8)^11).

(End)