A145882 -id:A145882 - cp's OEIS frontend

A168372 a(n) = n^5*(n^4 + 1)/2.

0, 1, 272, 9963, 131584, 978125, 5042736, 20185207, 67125248, 193739769, 500050000, 1179054371, 2580014592, 5302435333, 10330792304, 19222059375, 34360262656, 59294648177, 99180589968, 161345086939, 256001600000

Offset: 0

N. J. A. Sloane, Dec 11 2009

Number of unoriented rows of length 9 using up to n colors. For a(0)=0, there are no rows using no colors. For a(1)=1, there is one row using that one color for all positions. For a(2)=272, there are 2^9=512 oriented arrangements of two colors. Of these, 2^5=32 are achiral. That leaves (512-32)/2=240 chiral pairs. Adding achiral and chiral, we get 272. - Robert A. Russell, Nov 13 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (10, -45, 120, -210, 252, -210, 120, -45, 10, -1).

Row 9 of A277504.

Cf. A001017 (oriented), A000584 (achiral).

List([0..30], n -> n^5*(n^4 + 1)/2); # G. C. Greubel, Nov 15 2018

[n^5*(n^4+1)/2: n in [0..30]]; // Vincenzo Librandi, Aug 28 2011

Table[n^5*(n^4 + 1)/2, {n,0,50}] (* G. C. Greubel, Jul 19 2016 *)

a(n)=n^5*(n^4+1)/2 \\ Charles R Greathouse IV, Jul 19 2016

[n^5*(1 + n^4)/2 for n in range(30)] # G. C. Greubel, Nov 15 2018

G.f.: x*(1 + 262*x + 7288*x^2 + 44074*x^3 + 78190*x^4 + 44074*x^5 + 7288*x^6 + 262*x^7 + x^8)/(1 - x)^10. - G. C. Greubel, Jul 19 2016
From Robert A. Russell, Nov 13 2018: (Start)
a(n) = (A001017(n) + A000584(n)) / 2 = (n^9 + n^5) / 2.
G.f.: (Sum_{j=1..9} S2(9,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..5} S2(5,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.
G.f.: x*Sum_{k=0..8} A145882(9,k) * x^k / (1-x)^10.
E.g.f.: (Sum_{k=1..9} S2(9,k)*x^k + Sum_{k=1..5} S2(5,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.
For n>9, a(n) = Sum_{j=1..10} -binomial(j-11,j) * a(n-j). (End)
E.g.f.: x*(2 +270*x +3050*x^2 +7780*x^3 +6952*x^4 +2646*x^5 +462*x^6 + 36*x^7 +x^8)*exp(x)/2. - G. C. Greubel, Nov 15 2018

A168663 a(n) = n^7*(n^6 + 1)/2.

Original entry on oeis.org

0, 1, 4160, 798255, 33562624, 610390625, 6530486976, 48444916975, 274878955520, 1270935305649, 5000005000000, 17261365815551, 53496620605440, 151437584670385, 396857439333824, 973097619609375, 2251799947902976

Offset: 0

N. J. A. Sloane, Dec 11 2009

Number of unoriented rows of length 13 using up to n colors. For a(0)=0, there are no rows using no colors. For a(1)=1, there is one row using that one color for all positions. For a(2)=4160, there are 2^13=8192 oriented arrangements of two colors. Of these, 2^7=128 are achiral. That leaves (8192-128)/2=4032 chiral pairs. Adding achiral and chiral, we get 4160. - Robert A. Russell, Nov 13 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1).

Row 13 of A277504.

Cf. A010801 (oriented), A001015 (achiral).

[n^7*(n^6+1)/2: n in [0..20]]; // Vincenzo Librandi, Aug 28 2011

Table[n^7(n^6+1)/2,{n,0,20}] (* Harvey P. Dale, Jan 20 2013 *)

a(n)=n^7*(n^6+1)/2 \\ Charles R Greathouse IV, Jul 28 2016

From G. C. Greubel, Jul 28 2016: (Start)
G.f.: x*(1 + 4146*x + 740106*x^2 + 22765250*x^3 + 211641855*x^4 + 752814348*x^5 + 1137578988*x^6 + 752814348*x^7 + 211641855*x^8 + 22765250*x^9 + 740106*x^10 + 4146*x^11 + x^12)/(1 - x)^14.
E.g.f.: (1/2)*x*(2 + 4158*x + 261926*x^2 + 2532880*x^3 + 7508641*x^4 + 9321333*x^5 + 5715425*x^6 + 1899612*x^7 + 359502*x^8 + 39325*x^9 + 2431*x^10 + 78*x^11 + x^12)*exp(x). (End)
From Robert A. Russell, Nov 13 2018: (Start)
a(n) = (A010801(n) + A001015(n)) / 2 = (n^13 + n^7) / 2.
G.f.: (Sum_{j=1..13} S2(13,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..7} S2(7,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.
G.f.: x*Sum_{k=0..12} A145882(13,k) * x^k / (1-x)^14.
E.g.f.: (Sum_{k=1..13} S2(13,k)*x^k + Sum_{k=1..7} S2(7,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.
For n>13, a(n) = Sum_{j=1..14} -binomial(j-15,j) * a(n-j). (End)

A168664 a(n) = n^7*(n^7 + 1)/2.

Original entry on oeis.org

0, 1, 8256, 2392578, 134225920, 3051796875, 39182222016, 339111948196, 2199024304128, 11438398618965, 50000005000000, 189874926535206, 641959250190336, 1968688224223903, 5556003465485760, 14596463098125000, 36028797153181696, 84188913484869801

Offset: 0

N. J. A. Sloane, Dec 11 2009

Number of unoriented rows of length 14 using up to n colors. For a(0)=0, there are no rows using no colors. For a(1)=1, there is one row using that one color for all positions. For a(2)=8256, there are 2^14=16384 oriented arrangements of two colors. Of these, 2^7=128 are achiral. That leaves (16384-128)/2=8128 chiral pairs. Adding achiral and chiral, we get 8256. - Robert A. Russell, Nov 13 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (15,-105,455,-1365,3003,-5005,6435,-6435,5005, -3003,1365,-455,105,-15,1).

Cf. A001015 (Seventh Powers: n^7), A000217 (Triangular Numbers).
Row 14 of A277504.
Cf. A010802 (oriented), A001015 (achiral).

List([0..30], n -> n^7*(1 + n^7)/2); # G. C. Greubel, Nov 15 2018

[n^7*(n^7+1)/2: n in [0..20]]; // Vincenzo Librandi, Aug 28 2011

A168664:=n->n^7*(n^7+1)/2: seq(A168664(n), n=0..20); # Wesley Ivan Hurt, Oct 30 2014

f[n_]:=Module[{c=n^7},c (c+1)/2]; f/@Range[0,30] (* Harvey P. Dale, Mar 19 2011 *)

a(n)=n^7*(n^7+1)/2 \\ Charles R Greathouse IV, Jul 28 2016

[n^7*(1 + n^7)/2 for n in range(40)] # G. C. Greubel, Nov 15 2018

From Wesley Ivan Hurt, Oct 30 2014: (Start)
G.f.: (x + 8241*x^2 + 2268843*x^3 + 99203675*x^4 + 1285873650*x^5 + 6421633938*x^6 + 13985577438*x^7 + 13985598654*x^8 + 6421628925*x^9 + 1285868525*x^10 + 99207111*x^11 + 2268471*x^12 + 8128*x^13)/(1 - x)^15.
a(n) = 15*a(n-1) - 105*a(n-2) + 455*a(n-3) - 1365*a(n-4) + 3003*a(n-5) - 5005*a(n-6) + 6435*a(n-7) - 6435*a(n-8) + 5005*a(n-9) - 3003*a(n-10) + 1365*a(n-11) - 455*a(n-12) + 105*a(n-13) - 15*a(n-14) + a(n-15).
a(n) = n^7*(n^7 + 1)/2 = A000217(A001015(n)). (End)
From Robert A. Russell, Nov 13 2018: (Start)
a(n) = (A010802(n) + A001015(n)) / 2 = (n^14 + n^7) / 2.
G.f.: (Sum_{j=1..14} S2(14,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..7} S2(7,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.
G.f.: x*Sum_{k=0..13} A145882(14,k) * x^k / (1-x)^15.
E.g.f.: (Sum_{k=1..14} S2(14,k)*x^k + Sum_{k=1..7} S2(7,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.
For n>14, a(n) = Sum_{j=1..15} -binomial(j-16,j) * a(n-j). (End)
E.g.f.: x*(2+8254*x +789271*x^2 +10392095*x^3 +40075175*x^4 +63436394*x^5 +49329281*x^6 +20912320*x^7 +5135130*x^8 +752752*x^9 + 66066*x^10 +3367*x^11 +91*x^12 +x^13)*exp(x)/2. - G. C. Greubel, Nov 15 2018

A071235 a(n) = (n^12 + n^6)/2.

Original entry on oeis.org

0, 1, 2080, 266085, 8390656, 122078125, 1088414496, 6920702425, 34359869440, 141215033961, 500000500000, 1569215074141, 4458051717120, 11649044974645, 28346959952416, 64873174640625, 140737496743936, 291311130683665, 578415707719200, 1106657483056021

Offset: 0

N. J. A. Sloane, Jun 12 2002

Number of unoriented rows of length 12 using up to n colors. For a(0)=0, there are no rows using no colors. For a(1)=1, there is one row using that one color for all positions. For a(2)=2080, there are 2^12=4096 oriented arrangements of two colors. Of these, 2^6=64 are achiral. That leaves (4096-64)/2=2016 chiral pairs. Adding achiral and chiral, we get 2080. - Robert A. Russell, Nov 13 2018

T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (13, -78, 286, -715, 1287, -1716, 1716, -1287, 715, -286, 78, -13, 1).

Row 12 of A277504.

Cf. A008456 (oriented), A001014 (achiral).

List([0..40], n -> (n^12 + n^6)/2); # G. C. Greubel, Nov 15 2018

[n^6*(n^2+1)*(n^4-n^2+1)/2: n in [0..40]]; // Vincenzo Librandi, Jun 14 2011

Table[(n^12 + n^6)/2, {n,0,30}] (* Robert A. Russell, Nov 13 2018 *)

vector(40, n, n--; ) \\ G. C. Greubel, Nov 15 2018

for n in range(0,20): print(int((n**12 + n**6)/2), end=', ') # Stefano Spezia, Nov 15 2018

[n^6*(1 + n^6)/2 for n in range(40)] # G. C. Greubel, Nov 15 2018

a(n) = n^6*(n^2 + 1)*(n^4 - n^2 + 1)/2.
From Robert A. Russell, Nov 13 2018: (Start)
a(n) = (A008456(n) + A001014(n)) / 2 = (n^12 + n^6) / 2.
G.f.: (Sum_{j=1..12} S2(12,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..6} S2(6,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.
G.f.: x*Sum_{k=0..11} A145882(12,k) * x^k / (1-x)^13.
E.g.f.: (Sum_{k=1..12} S2(12,k)*x^k + Sum_{k=1..6} S2(6,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.
For n>12, a(n) = Sum_{j=1..13} -binomial(j-14,j) * a(n-j). (End)
From G. C. Greubel, Nov 15 2018: (Start)
G.f.: x*(1 +2067*x +239123*x^2 +5093505*x^3 +33160062*x^4 + 81255642*x^5 +81255642*x^6 +33160062*x^7 +5093505*x^8 +239123*x^9 +2067*x^10 +x^11)/( 1-x)^13.
E.g.f.: x*(2 +2078*x +86616*x^2 +611566*x^3 +1379415*x^4 +*1323653*x^5 + 627396*x^6 +159027*x^7 +22275*x^8 +1705*x^9 +66*x^10 +x^11)*exp(x)/2. (End)

New name from G. C. Greubel, Nov 15 2018

A168194 a(n) = n^4*(n^3 + 1)/2.

Original entry on oeis.org

0, 1, 72, 1134, 8320, 39375, 140616, 412972, 1050624, 2394765, 5005000, 9750906, 17926272, 31388539, 52725960, 85455000, 134250496, 205211097, 306162504, 447001030, 640080000, 900641511, 1247296072, 1702552644, 2293401600

Offset: 0

N. J. A. Sloane, Dec 11 2009

Number of unoriented rows of length 7 using up to n colors. For a(0)=0, there are no rows using no colors. For a(1)=1, there is one row using that one color for all positions. For a(2)=72, there are 2^7=128 oriented arrangements of two colors. Of these, 2^4=16 are achiral. That leaves (128-16)/2=56 chiral pairs. Adding achiral and chiral, we get 72. - Robert A. Russell, Nov 13 2018

Vincenzo Librandi and Bruno Berselli, Table of n, a(n) for n = 0..1000 (first 480 terms from Vincenzo Librandi).
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Row 7 of A277504.

Cf. A001015 (oriented), A000583 (achiral).

[n^4*(n^3+1)/2: n in [0..50]]; // Vincenzo Librandi, Apr 25 2011

Table[(n^4 (n^3+1))/2,{n,0,40}] (* Harvey P. Dale, Apr 29 2011 *)

vector(50, n, n--; n^4*(n^3+1)/2) \\ G. C. Greubel, Nov 14 2018

[n^4*(n^3+1)/2 for n in (0..50)] # G. C. Greubel, Nov 14 2018

G.f.: x*(1 + 64*x + 586*x^2 + 1208*x^3 + 605*x^4 + 56*x^5)/(1-x)^8. - Colin Barker, Apr 26 2012
From Robert A. Russell, Nov 13 2018: (Start)
a(n) = (A001015(n) + A000583(n)) / 2 = (n^7 + n^4) / 2.
G.f.: (Sum_{j=1..7} S2(7,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..4} S2(4,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.
G.f.: x*Sum_{k=0..6} A145882(7,k) * x^k / (1-x)^8.
E.g.f.: (Sum_{k=1..7} S2(7,k)*x^k + Sum_{k=1..4} S2(4,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.
For n>7, a(n) = Sum_{j=1..8} -binomial(j-9,j) * a(n-j). (End)
E.g.f.: x*(2 +70*x +307*x^2 +351*x^3 +140*x^4 +21*x^5 +x^6)*exp(x)/2. - G. C. Greubel, Nov 14 2018

A168627 a(n) = n^6*(n^5 + 1)/2.

Original entry on oeis.org

0, 1, 1056, 88938, 2099200, 24421875, 181421856, 988722196, 4295098368, 15690795525, 50000500000, 142656721086, 371505678336, 896082610423, 2024786349600, 4324883625000, 8796101410816, 17135960222601

Offset: 0

N. J. A. Sloane, Dec 11 2009

nonn

Number of unoriented rows of length 11 using up to n colors. For a(0)=0, there are no rows using no colors. For a(1)=1, there is one row using that one color for all positions. For a(2)=1056, there are 2^11=2048 oriented arrangements of two colors. Of these, 2^6=64 are achiral. That leaves (2048-64)/2=992 chiral pairs. Adding achiral and chiral, we get 1056. - Robert A. Russell, Nov 13 2018

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12, -66, 220, -495, 792, -924, 792, -495, 220, -66, 12, -1).

Row 11 of A277504.

Cf. A008455 (oriented), A001014 (achiral).

List([0..30], n -> n^6*(1 + n^5)/2); # G. C. Greubel, Nov 15 2018

[n^6*(1 + n^5)/2: n in [0..30]]; // G. C. Greubel, Nov 15 2018

Table[n^6*(n^5+1)/2, {n, 0, 30}] (* G. C. Greubel, Jul 27 2016 *)
LinearRecurrence[{12,-66,220,-495,792,-924,792,-495,220,-66,12,-1},{0,1,1056,88938,2099200,24421875,181421856,988722196,4295098368,15690795525,50000500000,142656721086},20] (* Harvey P. Dale, Nov 21 2024 *)

vector(30, n, n--; n^6*(1 + n^5)/2) \\ G. C. Greubel, Nov 15 2018

[n^6*(1 + n^5)/2 for n in range(30)] # G. C. Greubel, Nov 15 2018

From G. C. Greubel, Jul 27 2016: (Start)
G.f.: x*(1 + 1044*x + 76332*x^2 + 1101420*x^3 + 4869558*x^4 + 7862124*x^5 + 4868556*x^6 + 1102068*x^7 + 76305*x^8 + 992*x^9)/(1 - x)^12.
E.g.f.: (1/2)* x *(2 + 1054*x + 28591*x^2 + 145815*x^3 + 246745*x^4 + 179488*x^5 + 63987*x^6 + 11880*x^7 + 1155*x^8 + 55*x^9 + x^10)*exp(x). (End)
From Robert A. Russell, Nov 13 2018: (Start)
a(n) = (A008455(n) + A001014(n)) / 2 = (n^11 + n^6) / 2.
G.f.: (Sum_{j=1..11} S2(11,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..6} S2(6,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.
G.f.: x*Sum_{k=0..10} A145882(11,k) * x^k / (1-x)^12.
E.g.f.: (Sum_{k=1..11} S2(11,k)*x^k + Sum_{k=1..6} S2(6,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.
For n>11, a(n) = Sum_{j=1..12} -binomial(j-13,j) * a(n-j). (End)

A170779 a(n) = n^8*(n^7 + 1)/2.

Original entry on oeis.org

0, 1, 16512, 7177734, 536903680, 15258984375, 235093332096, 2373783637372, 17592194433024, 102945587570685, 500000050000000, 2088624191887266, 7703511002284032, 25592946914910739, 77784048516800640

Offset: 0

N. J. A. Sloane, Dec 11 2009

Number of unoriented rows of length 15 using up to n colors. For a(0)=0, there are no rows using no colors. For a(1)=1, there is one row using that one color for all positions. For a(2)=16512, there are 2^15=32768 oriented arrangements of two colors. Of these, 2^8=256 are achiral. That leaves (32768-256)/2=16256 chiral pairs. Adding achiral and chiral, we get 16512. - Robert A. Russell, Nov 13 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (16, -120, 560, -1820, 4368, -8008, 11440, -12870, 11440, -8008, 4368, -1820, 560, -120, 16, -1).

Row 15 of A277504.

Cf. A010803 (oriented), A001016 (achiral).

[n^8*(n^7+1)/2: n in [0..30]]; // Vincenzo Librandi, Aug 26 2011

Table[n^8*(n^7+1)/2, {n,0,30}] (* G. C. Greubel, Dec 05 2017 *)

for(n=0, 30, print1(n^8*(n^7+1)/2, ", ")) \\ G. C. Greubel, Dec 05 2017

[n^8*(n^7+1)/2 for n in range(30)] # G. C. Greubel, Nov 15 2018

G.f.: (x + 16496*x^2 + 6913662*x^3 + 424040816*x^4 + 7520608675*x^5 + 51388540128*x^6 + 155693747508*x^7 + 223769408736*x^8 + 155693850903*x^9 + 51388458800*x^10 + 7520620846*x^11 + 424050096*x^12 + 6911077*x^13 + 16256*x^14)/(1-x)^16. - G. C. Greubel, Dec 05 2017
From Robert A. Russell, Nov 13 2018: (Start)
a(n) = (A010803(n) + A001016(n)) / 2 = (n^15 + n^8) / 2.
G.f.: (Sum_{j=1..15} S2(15,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..8} S2(8,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.
G.f.: x*Sum_{k=0..14} A145882(15,k) * x^k / (1-x)^16.
E.g.f.: (Sum_{k=1..15} S2(15,k)*x^k + Sum_{k=1..8} S2(8,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.
For n>15, a(n) = Sum_{j=1..16} -binomial(j-17,j) * a(n-j). (End)
E.g.f.: x*(2 +16510*x +2376067*x^2 +42357651*x^3 +210767970*x^4 + 420693539*x^5 +408741361*x^6 +216627841*x^7 +67128490*x^8 + 12662650*x^9 +1479478*x^10 +106470*x^11 +4550*x^12 +105*x^13 +x^14)*exp(x)/2. - G. C. Greubel, Nov 15 2018

A170780 a(n) = n^8*(n^8 + 1)/2.

Original entry on oeis.org

0, 1, 32896, 21526641, 2147516416, 76294140625, 1410555793536, 16616468167201, 140737496743936, 926510115949281, 5000000050000000, 22974865038965521, 92442129662509056, 332708304999455281, 1088976669642580096

Offset: 0

N. J. A. Sloane, Dec 11 2009

Number of unoriented rows of length 16 using up to n colors. For a(0)=0, there are no rows using no colors. For a(1)=1, there is one row using that one color for all positions. For a(2)=32896, there are 2^16=65536 oriented arrangements of two colors. Of these, 2^8=256 are achiral. That leaves (65536-256)/2=32640 chiral pairs. Adding achiral and chiral, we get 32896. - Robert A. Russell, Nov 13 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (17, -136, 680, -2380, 6188, -12376, 19448, -24310, 24310, -19448, 12376, -6188, 2380, -680, 136, -17, 1).

Row 16 of A277504.

Cf. A010804 (oriented), A001016 (achiral).

List([0..30], n -> n^8*(n^8+1)/2); # G. C. Greubel, Nov 15 2018

[n^8*(n^8+1)/2: n in [0..30]]; // Vincenzo Librandi, Aug 26 2011

Table[n^8*(n^8+1)/2, {n, 0, 30}] (* G. C. Greubel, Dec 05 2017 *)

for(n=0, 30, print1(n^8*(n^8+1)/2, ", ")) \\ G. C. Greubel, Dec 05 2017

for n in range(0,20): print(int(n**8*(n**8 + 1)/2), end=', ') # Stefano Spezia, Nov 15 2018

[n^8*(n^8+1)/2 for n in range(30)] # G. C. Greubel, Nov 15 2018

G.f.: (x + 32879*x^2 + 20967545*x^3 + 1786036695*x^4 + 42691617829* x^5 + 391057805899*x^6 + 1603741496717*x^7 + 3191399514435*x^8 + 3191399514435*x^9 + 1603741496717*x^10 + 391057805899*x^11 + 42691617829*x^12 + 1786036695*x^13 + 20967545*x^14 + 32879*x^15 + x^16) /(1-x)^17. - G. C. Greubel, Dec 05 2017
From Robert A. Russell, Nov 13 2018: (Start)
a(n) = (A010804(n) + A001016(n)) / 2 = (n^16 + n^8) / 2.
G.f.: (Sum_{j=1..16} S2(16,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..8} S2(8,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.
G.f.: x*Sum_{k=0..15} A145882(16,k) * x^k / (1-x)^17.
E.g.f.: (Sum_{k=1..16} S2(16,k)*x^k + Sum_{k=1..8} S2(8,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.
For n>16, a(n) = Sum_{j=1..17} -binomial(j-18,j) * a(n-j). (End)

A170790 a(n) = n^9*(n^8 + 1)/2.

Original entry on oeis.org

0, 1, 65792, 64579923, 8590065664, 381470703125, 8463334761216, 116315277170407, 1125899973951488, 8338591043543529, 50000000500000000, 252723515428620731, 1109305555950108672, 4325207964992918653

Offset: 0

N. J. A. Sloane, Dec 11 2009

Number of unoriented rows of length 17 using up to n colors. For a(0)=0, there are no rows using no colors. For a(1)=1, there is one row using that one color for all positions. For a(2)=65792, there are 2^17=131072 oriented arrangements of two colors. Of these, 2^9=512 are achiral. That leaves (131072-512)/2=65280 chiral pairs. Adding achiral and chiral, we get 65792. - Robert A. Russell, Nov 13 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (18,-153,816,-3060,8568,-18564, 31824,-43758,48620,-43758,31824,-18564,8568,-3060,816,-153,18,-1).

Row 17 of A277504.

Cf. A010805 (oriented), A001017 (achiral).

List([0..30], n -> n^9*(n^8+1)/2); # G. C. Greubel, Nov 15 2018

[n^9*(n^8+1)/2: n in [0..20]]; // Vincenzo Librandi, Aug 26 2011

Table[(n^9 (n^8+1))/2,{n,0,20}] (* Harvey P. Dale, Oct 03 2016 *)

for(n=0,30, print1(n^9*(n^8+1)/2, ", ")) \\ G. C. Greubel, Dec 06 2017

for n in range(0,20): print(int(n**9*(n**8 + 1)/2), end=', ') # Stefano Spezia, Nov 15 2018

[n^9*(n^8+1)/2 for n in range(30)] # G. C. Greubel, Nov 15 2018

G.f.: (x + 65774*x^2 + 63395820*x^3 + 7437692410*x^4 + 236676566180*x^5 + 2858646249342*x^6 + 15527826341908*x^7 + 41568611082650*x^8 + 57445191259830*x^9 + 41568611082650*x^10 + 15527826341908*x^11 + 2858646249342*x^12 + 236676566180*x^13 + 7437692410*x^14 + 63395820*x^15 + 65774*x^16 + x^17)/(1-x)^18. - G. C. Greubel, Dec 06 2017
From Robert A. Russell, Nov 13 2018: (Start)
a(n) = (A010805(n) + A001017(n)) / 2 = (n^17 + n^9) / 2.
G.f.: (Sum_{j=1..17} S2(17,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..9} S2(9,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.
G.f.: x*Sum_{k=0..16} A145882(17,k) * x^k / (1-x)^18.
E.g.f.: (Sum_{k=1..17} S2(17,k)*x^k + Sum_{k=1..9} S2(9,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.
For n>17, a(n) = Sum_{j=1..18} -binomial(j-19,j) * a(n-j). (End)

A170791 a(n) = n^9*(n^9 + 1)/2.

Original entry on oeis.org

0, 1, 131328, 193720086, 34359869440, 1907349609375, 50779983373056, 814206819132028, 9007199321849856, 75047317842209805, 500000000500000000, 2779958657925089586, 13311666643022512128, 56227703481280946251

Offset: 0

N. J. A. Sloane, Dec 11 2009

Number of unoriented rows of length 18 using up to n colors. For a(0)=0, there are no rows using no colors. For a(1)=1, there is one row using that one color for all positions. For a(2)=131328, there are 2^18=262144 oriented arrangements of two colors. Of these, 2^9=512 are achiral. That leaves (262144-512)/2=130816 chiral pairs. Adding achiral and chiral, we get 131328. - Robert A. Russell, Nov 13 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (19,-171,969,-3876,11628,-27132, 50388,-75582,92378,-92378,75582,-50388,27132,-11628,3876,-969,171, -19,1).

Row 18 of A277504.

Cf. A010806 (oriented), A001017 (achiral).

List([0..30], n -> n^9*(n^9 + 1)/2); # G. C. Greubel, Nov 15 2018

[n^9*(n^9+1)/2: n in [0..20]]; // Vincenzo Librandi, Aug 26 2011

f[n_]:=Module[{n9=n^9},(n9(n9+1))/2]; Array[f,20,0] (* Harvey P. Dale, Nov 24 2012 *)
Table[n^9*(n^9+1)/2, {n,0,30}] (* G. C. Greubel, Dec 06 2017 *)

for(n=0,30, print1(n^9*(n^9+1)/2, ", ")) \\ G. C. Greubel, Dec 06 2017

for n in range(0,20): print(int(n**9*(n**9 + 1)/2), end=', ') # Stefano Spezia, Nov 15 2018

[n^9*(1 + n^9)/2 for n in range(30)] # G. C. Greubel, Nov 15 2018

G.f.: (x + 131309*x^2 + 191225025*x^3 + 30701643925*x^4 + 1287510971765*x^5 + 20228672721537*x^6 + 142998536758213*x^7 + 503354983579865*x^8 + 932692830330915*x^9 + 932692827449735*x^10 + 503354984335363*x^11 + 142998537549087*x^12 + 20228672026535*x^13 + 1287511125835*x^14 + 30701669175*x^15 + 191214899*x^16 + 130816*x^17) /(1-x)^19. - G. C. Greubel, Dec 06 2017
From Robert A. Russell, Nov 13 2018: (Start)
a(n) = (A010806(n) + A001017(n)) / 2 = (n^18 + n^9) / 2.
G.f.: (Sum_{j=1..18} S2(18,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..9} S2(9,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.
G.f.: x*Sum_{k=0..17} A145882(18,k) * x^k / (1-x)^19.
E.g.f.: (Sum_{k=1..18} S2(18,k)*x^k + Sum_{k=1..9} S2(9,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.
For n>18, a(n) = Sum_{j=1..19} -binomial(j-20,j) * a(n-j). (End)

cp's OEIS Frontend

A168372 a(n) = n^5*(n^4 + 1)/2.

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A168664 a(n) = n^7*(n^7 + 1)/2.

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A168194 a(n) = n^4*(n^3 + 1)/2.

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A168627 a(n) = n^6*(n^5 + 1)/2.

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