A170798 -id:A170798 - cp's OEIS frontend

A170801 a(n) = n^10*(n^9 + 1)/2.

0, 1, 262656, 581160258, 137439477760, 9536748046875, 304679900238336, 5699447733924196, 72057594574798848, 675425860579888245, 5000000005000000000, 30579545237175985446, 159739999716270145536

Offset: 0

N. J. A. Sloane, Dec 11 2009

Number of unoriented rows of length 19 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)=262656, there are 2^19=524288 oriented arrangements of two colors. Of these, 2^10=1024 are achiral. That leaves (524288-1024)/2=261632 chiral pairs. Adding achiral and chiral, we get 262656. - 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 (20, -190, 1140, -4845, 15504, -38760, 77520, -125970, 167960, -184756, 167960, -125970, 77520, -38760, 15504, -4845, 1140, -190, 20, -1).

Row 19 of A277504.
Cf. A010807 (oriented), A008454 (achiral).
Sequences of the form n^10*(n^m + 1)/2: A170793 (m=1), A170794 (m=2), A170795 (m=3), A170796 (m=4), A170797 (m=5), A170798 (m=6), A170799 (m=7), A170800 (m=8), this sequence (m=9), A170802 (m=10).

List([0..30], n -> n^10*(n^9+1)/2); # G. C. Greubel, Nov 15 2018

[n^10*(n^9+1)/2: n in [0..20]]; // Vincenzo Librandi, Aug 27 2011

seq(n^10*(n^9 +1)/2, n=0..20); # G. C. Greubel, Oct 11 2019

Table[(n^19 + n^10)/2, {n,0,30}] (* Robert A. Russell, Nov 13 2018 *)

vector(30, n, n--; n^10*(n^9+1)/2) \\ G. C. Greubel, Nov 15 2018

[n^10*(n^9+1)/2 for n in range(30)] # G. C. Greubel, Nov 15 2018

From Robert A. Russell, Nov 13 2018: (Start)
a(n) = (A010807(n) + A008454(n)) / 2 = (n^19 + n^10) / 2.
G.f.: (Sum_{j=1..19} S2(19,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..10} S2(10,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.
G.f.: x*Sum_{k=0..18} A145882(19,k) * x^k / (1-x)^20.
E.g.f.: (Sum_{k=1..19} S2(19,k)*x^k + Sum_{k=1..10} S2(10,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.
For n>19, a(n) = Sum_{j=1..20} -binomial(j-21,j) * a(n-j). (End)

A170802 a(n) = n^10*(n^10 + 1)/2.

Original entry on oeis.org

0, 1, 524800, 1743421725, 549756338176, 47683720703125, 1828079250264576, 39896133290043625, 576460752840294400, 6078832731271856601, 50000000005000000000, 336374997479248716901, 1916879996254696243200

Offset: 0

N. J. A. Sloane, Dec 11 2009

By definition, all terms are triangular numbers. - Harvey P. Dale, Aug 12 2012

Number of unoriented rows of length 20 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)=524800, there are 2^20=1048576 oriented arrangements of two colors. Of these, 2^10=1024 are achiral. That leaves (1048576-1024)/2=523776 chiral pairs. Adding achiral and chiral, we get 524800. - 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 (21, -210, 1330, -5985, 20349, -54264, 116280, -203490, 293930, -352716, 352716, -293930, 203490, -116280, 54264, -20349, 5985, -1330, 210, -21, 1).

Row 20 of A277504.
Cf. A010808 (oriented), A008454 (achiral).
Sequences of the form n^10*(n^m + 1)/2: A170793 (m=1), A170794 (m=2), A170795 (m=3), A170896 (m=4), A170797 (m=5), A170798 (m=6), A170799 (m=7), A170800 (m=8), A170801 (m=9), this sequence (m=10).

List([0..30], n -> n^10*(n^10+1)/2); # G. C. Greubel, Nov 15 2018

[n^10*(n^10+1)/2: n in [0..20]]; // Vincenzo Librandi, Aug 27 2011

seq(n^10*(n^10 +1)/2, n=0..20); # G. C. Greubel, Oct 11 2019

n10[n_]:=Module[{c=n^10},(c(c+1))/2];Array[n10,15,0] (* Harvey P. Dale, Jul 17 2012 *)

vector(30, n, n--; n^10*(n^10+1)/2) \\ G. C. Greubel, Nov 15 2018

for n in range(0,20): print(int(n**10*(n**10 + 1)/2), end=', ') # Stefano Spezia, Nov 15 2018

[n^10*(n^10+1)/2 for n in range(30)] # G. C. Greubel, Nov 15 2018

From Robert A. Russell, Nov 13 2018: (Start)
a(n) = (A010808(n) + A008454(n)) / 2 = (n^20 + n^10) / 2.
G.f.: (Sum_{j=1..20} S2(20,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..10} S2(10,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.
G.f.: x*Sum_{k=0..19} A145882(20,k) * x^k / (1-x)^21.
E.g.f.: (Sum_{k=1..20} S2(20,k)*x^k + Sum_{k=1..10} S2(10,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.
For n>20, a(n) = Sum_{j=1..21} -binomial(j-22,j) * a(n-j). (End)

A283029 Number of inequivalent 5 X 5 matrices with entries in {1,2,3,..,n} when a matrix and its transpose are considered equivalent.

Original entry on oeis.org

0, 1, 16793600, 423651479175, 562950490292224, 149011627197265625, 14215144250057342976, 670534312205763205375, 18889465949070766899200, 358948993948871860432449, 5000000000500000000000000, 54173529719030485105622951, 476981083228048575587942400

Offset: 0

David Nacin, Feb 27 2017

Cycle index of symmetric group S2 acting on the set of 25 entries is (s(2)^10*s(1)^5 + s(1)^25)/2.

			For n=2 we get a(2)=16793600 inequivalent 5x5 binary matrices up to the action of transposition.

Cf. A282612,A282613,A282614. A283026, A283027, A283028, A283030, A283031, A283032, A283033. A170798 (4x4 version). A168555 (3x3 version). A019582 (2x2 version)

Table[n^15 (n^2 + 1) (n^8 - n^6 + n^4 - n^2 + 1)/2, {n, 0, 12}]

a(n) = n^15*(n^2+1)*(n^8-n^6+n^4-n^2+1)/2; \\ Indranil Ghosh, Feb 27 2017

def A283029(n): return n**15*(n**2+1)*(n**8-n**6+n**4-n**2+1)/2 # Indranil Ghosh, Feb 27 2017

a(n) = n^15*(n^2+1)*(n^8-n^6+n^4-n^2+1)/2.
From Chai Wah Wu, Dec 07 2018: (Start)
a(n) = 26*a(n-1) - 325*a(n-2) + 2600*a(n-3) - 14950*a(n-4) + 65780*a(n-5) - 230230*a(n-6) + 657800*a(n-7) - 1562275*a(n-8) + 3124550*a(n-9) - 5311735*a(n-10) + 7726160*a(n-11) - 9657700*a(n-12) + 10400600*a(n-13) - 9657700*a(n-14) + 7726160*a(n-15) - 5311735*a(n-16) + 3124550*a(n-17) - 1562275*a(n-18) + 657800*a(n-19) - 230230*a(n-20) + 65780*a(n-21) - 14950*a(n-22) + 2600*a(n-23) - 325*a(n-24) + 26*a(n-25) - a(n-26) for n > 25.
G.f.: x*(x^24 + 16793574*x^23 + 423214845900*x^22 + 551941009751074*x^21 + 134512557517054626*x^20 + 10522699609491808746*x^19 + 347912001753554722204*x^18 + 5696453728178627889150*x^17 + 50977946159336791604079*x^16 + 265857130683340877431996*x^15 + 842694350441988138095256*x^14 + 1667306282568523129263444*x^13 + 2089823554970188253479900*x^12 + 1667306282568523129263444*x^11 + 842694350441988138095256*x^10 + 265857130683340877431996*x^9 + 50977946159336791604079*x^8 + 5696453728178627889150*x^7 + 347912001753554722204*x^6 + 10522699609491808746*x^5 + 134512557517054626*x^4 + 551941009751074*x^3 + 423214845900*x^2 + 16793574*x + 1)/(x - 1)^26. (End)

A170797 a(n) = n^10*(n^5+1)/2.

Original entry on oeis.org

0, 1, 16896, 7203978, 537395200, 15263671875, 235122725376, 2373921992596, 17592722915328, 102947309439525, 500005000000000, 2088637053420126, 7703541745975296, 25593015436291303, 77784192406233600

Offset: 0

N. J. A. Sloane, Dec 11 2009

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).

Sequences of the form n^10*(n^m + 1)/2: A170793 (m=1), A170794 (m=2), A170795 (m=3), A170796 (m=4), this sequence (m=5), A170798 (m=6), A170799 (m=7), A170800 (m=8), A170801 (m=9), A170802 (m=10).

List([0..20], n-> n^10*(n^5 +1)/2); # G. C. Greubel, Oct 11 2019

[n^10*(n^5+1)/2: n in [0..20]]; // Vincenzo Librandi, Aug 26 2011

A170797:=n->n^10*(n^5+1)/2: seq(A170797(n), n=0..20); # Wesley Ivan Hurt, Aug 10 2016

Table[n^10*(n^5 + 1)/2, {n, 0, 15}] (* Wesley Ivan Hurt, Aug 10 2016 *)

vector(21, m, (m-1)^10*((m-1)^5 + 1)/2) \\ G. C. Greubel, Oct 11 2019

[n^10*(n^5 +1)/2 for n in (0..20)] # G. C. Greubel, Oct 11 2019

G.f.: x*(15872*x^13 +6890977*x^12 +423932400*x^11 +7520863426*x^10 +51389080880*x^9 +155692452591*x^8 +223769408736*x^7 +155695145820*x^6 +51387918048*x^5 +7520366095*x^4 +424158512*x^3 +6933762*x^2 +16880*x +1) / (x-1)^16. - Colin Barker, Nov 01 2014
a(n) = 16*a(n-1) - 120*a(n-2) + 560*a(n-3) - 1820*a(n-4) + 4368*a(n-5) - 8008*a(n-6) + 11440*a(n-7) - 12870*a(n-8) + 11440*a(n-9) - 8008*a(n-10) + 4368*a(n-11) - 1820*a(n-12) + 560*a(n-13) - 120*a(n-14) + 16*a(n-15) - a(n-16) for n > 15. - Wesley Ivan Hurt, Aug 10 2016
E.g.f.: x*(2 +16894*x +2384431*x^2 +42390055*x^3 +210809445*x^4 + 420716100*x^5 +408747213*x^6 +216628590*x^7 +67128535*x^8 +12662651*x^9 +1479478*x^10 +106470*x^11 +4550*x^12 +105*x^13 +x^14)*exp(x)/2. - G. C. Greubel, Oct 11 2019

A170799 a(n) = n^10*(n^7 + 1)/2.

Original entry on oeis.org

0, 1, 66048, 64599606, 8590458880, 381474609375, 8463359955456, 116315398231228, 1125900443713536, 8338592593225485, 50000005000000000, 252723527218359186, 1109305584328900608, 4325208028619914891, 15245673509292925440, 49263062956171875000, 147573953139432226816

Offset: 0

N. J. A. Sloane, Dec 11 2009

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).

Sequences of the form n^10*(n^m + 1)/2: A170793 (m=1), A170794 (m=2), A170795 (m=3), A170796 (m=4), A170797 (m=5), A170798 (m=6), this sequence (m=7), A170800 (m=8), A170801 (m=9), A170802 (m=10).

List([0..20], n-> n^10*(n^7 +1)/2); # G. C. Greubel, Oct 11 2019

[n^10*(n^7+1)/2: n in [0..20]]; // Vincenzo Librandi, Aug 27 2011

seq(n^10*(n^7 +1)/2, n=0..20); # G. C. Greubel, Oct 11 2019

Table[n^10(n^7+1)/2,{n,0,20}] (* Harvey P. Dale, Aug 27 2013 *)

vector(21, m, (m-1)^10*((m-1)^7 + 1)/2) \\ G. C. Greubel, Oct 11 2019

[n^10*(n^7 +1)/2 for n in (0..20)] # G. C. Greubel, Oct 11 2019

G.f.: x*(65024*x^15 + 63370125*x^14 + 7437628950*x^13 + 236677103915*x^12 + 2858645957220*x^11 + 15527824213413*x^10 + 41568614867330*x^9 + 57445190329275*x^8 + 41568608318040*x^7 + 15527828734975*x^6 + 2858646015162*x^5 + 236676197145*x^4 + 7437770500*x^3 + 63410895*x^2 + 66030*x + 1)/(x-1)^18. - Colin Barker, Feb 24 2013

E.g.f.: x*(2 + 66046*x + 21467155*x^2 + 694371395*x^3 + 5652794176*x^4 + 17505772725*x^5 + 25708110666*x^6 + 20415995778*x^7 + 9528822348*x^8 + 2758334151*x^9 + 512060978*x^10 + 62022324*x^11 + 4910178*x^12 + 249900*x^13 + 7820*x^14 + 136*x^15 + x^16)*exp(x)/2. - G. C. Greubel, Oct 12 2019

A170800 a(n) = n^10*(n^8 + 1)/2.

Original entry on oeis.org

0, 1, 131584, 193739769, 34360262656, 1907353515625, 50780008567296, 814206940192849, 9007199791611904, 75047319391891761, 500000005000000000, 2779958669714828041, 13311666671401304064, 56227703544907942489

Offset: 0

N. J. A. Sloane, Dec 11 2009

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).

Sequences of the form n^10*(n^m + 1)/2: A170793 (m=1), A170794 (m=2), A170795 (m=3), A170796 (m=4), A170797 (m=5), A170798 (m=6), A170799 (m=7), this sequence (m=8), A170801 (m=9), A170802 (m=10).

List([0..20], n-> n^10*(n^8 +1)/2); # G. C. Greubel, Oct 11 2019

[n^10*(n^8+1)/2: n in [0..20]]; // Vincenzo Librandi, Aug 27 2011

seq(n^10*(n^8 +1)/2, n=0..20); # G. C. Greubel, Oct 11 2019

Table[n^10 (n^8+1)/2,{n,0,20}] (* Harvey P. Dale, Jul 14 2013 *)

vector(21, m, (m-1)^10*((m-1)^8 + 1)/2) \\ G. C. Greubel, Oct 11 2019

[n^10*(n^8 +1)/2 for n in (0..20)] # G. C. Greubel, Oct 11 2019

G.f.: x*(1 + 131565*x + 191239844*x^2 + 30701706940*x^3 + 1287510524640*x^4 + 20228672856392*x^5 + 142998539385460*x^6 + 503354978422188*x^7 + 932692832164970*x^8 + 932692832164970*x^9 + 503354978422188*x^10 + 142998539385460*x^11 + 20228672856392*x^12 + 1287510524640*x^13 + 30701706940*x^14 +191239844*x^15 + 131565*x^16 + x^17)/(1-x)^19. - Harvey P. Dale, Jul 14 2013

E.g.f.: x*(2 + 131582*x + 64448340*x^2 + 2798841090*x^3 + 28958138070*x^4 + 110687273866*x^5 + 197462489280*x^6 + 189036065760*x^7 + 106175395800*x^8 + 37112163804*x^9 + 8391004908*x^10 + 1256328866*x^11 + 125854638*x^12 + 8408778*x^13 + 367200*x^14 + 9996*x^15 + 153*x^16 + x^17)*exp(x)/2. - G. C. Greubel, Oct 12 2019

A170796 a(n) = n^10*(n^4 + 1)/2.

Original entry on oeis.org

0, 1, 8704, 2421009, 134742016, 3056640625, 39212315136, 339252774049, 2199560126464, 11440139619681, 50005000000000, 189887885503921, 641990190956544, 1968757122095569, 5556148040106496, 14596751337890625

Offset: 0

N. J. A. Sloane, Dec 11 2009

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).

Sequences of the form n^10*(n^m + 1)/2: A170793 (m=1), A170794 (m=2), A170795 (m=3), this sequence (m=4), A170797 (m=5), A170798 (m=6), A170799 (m=7), A170800 (m=8), A170801 (m=9), A170802 (m=10).

List([0..20], n-> n^10*(n^4 +1)/2); # G. C. Greubel, Oct 11 2019

[n^10*(n^4+1)/2: n in [0..20]]; // Vincenzo Librandi, Aug 26 2011

seq(n^10*(n^4 +1)/2, n=0..20); # G. C. Greubel, Oct 11 2019

Table[n^10*(n^4 +1)/2, {n,0,20}] (* G. C. Greubel, Oct 11 2019 *)

vector(21, m, (m-1)^10*((m-1)^4 + 1)/2) \\ G. C. Greubel, Oct 11 2019

[n^10*(n^4 +1)/2 for n in (0..20)] # G. C. Greubel, Oct 11 2019

From G. C. Greubel, Oct 11 2019: (Start)
G.f.: x*(1 +8689*x +2290554*x^2 +99340346*x^3 +1285757375*x^4 +6420936303*x^5 +13986239532*x^6 +13986239532*x^7 +6420936303*x^8 +1285757375*x^9 +99340346*x^10 +2290554*x^11 +8689*x^12 +x^13)/(1-x)^15.
E.g.f.: x*(2 +8702*x +798300*x^2 +10425850*x^3 +40117560*x^4 +63459200*x^5 +49335160*x^6 +20913070*x^7 +5135175*x^8 +752753*x^9 + 66066*x^10 +3367*x^11 +91*x^12 +x^13)*exp(x)/2. (End)

cp's OEIS Frontend

A170801 a(n) = n^10*(n^9 + 1)/2.

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A170802 a(n) = n^10*(n^10 + 1)/2.

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A283029 Number of inequivalent 5 X 5 matrices with entries in {1,2,3,..,n} when a matrix and its transpose are considered equivalent.

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A170797 a(n) = n^10*(n^5+1)/2.

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A170799 a(n) = n^10*(n^7 + 1)/2.

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A170800 a(n) = n^10*(n^8 + 1)/2.

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