Number of bracelets (COMMENT of sequence A001399)

Fri Jul 11 12:45:52 CEST 2008

It is here:

http://web.archive.org/web/20040714202813/http://www.kent.ac.uk/ims/publications/documents/paper_607.pdf

http://web.archive.org  is worth trying for all dead links.

Brendan.

* N. J. A. Sloane <njas at research.att.com> [080711 22:47]:
> Dear Seqfans,  can someone figure out what the definition means?
> The link is broken, and A N W Hone has not responded to
> my asking for a new URL for the article.
> Neil
> 
> %I A122046
> %S A122046 0,0,1,3,6,10,16,24,34,46,61,79,100,124,152,184,220,260,305,355,410
> %N A122046 a(n)=(x^(n - 1)a(n - 1)a(n - 4) + a(n - 2)*a(n - 3))/a(n - 5).
> %F A122046 Comment from Jerry Metzger (jerry_metzger(AT)und.nodak.edu), Jul 09 2008: It appears that every 4th term is given by Sum_{m=1...n-1} (floor{m(n-2)/n})^2 and the intermediate terms by adding an easy "adjustment" term to that sum.
> %H A122046 A. N. W. Hone, <a href="http://www.kent.ac.uk/ims/publications/documents/paper_607.pdf">Algebraic curves, integer sequences and a discrete Painleve transcendent</a>, Proceedings of SIDE 6, Helsinki, Finland, Jun 19 2004. [Set a(n)=d(n+3) on p. 8]
> %t A122046 p[n_] := p[n] = Cancel[Simplify[ (x^(n - 1)p[n - 1]p[n - 4] + p[n - 2]*p[n - 3])/p[n - 5]]]; p[ -5] = 1;p[ -4] = 1;p[ -3] = 1;p[ -2] = 1;p[ -1] = 1; Table[Exponent[p[n], x], {n, 0, 20}]
> %Y A122046 Cf. A014125.
> %Y A122046 Sequence in context: A121776 A088637 A066377 this_sequence A078663 A025222 A011902
> %Y A122046 Adjacent sequences: A122043 A122044 A122045 this_sequence A122047 A122048 A122049
> %K A122046 nonn,uned,obsc
> %O A122046 0,4
> %A A122046 Roger Bagula (rlbagulatftn(AT)yahoo.com), Sep 13 2006
> %E A122046 Partially edited by njas, Sep 17 2006, Jul 11 2008
> 

njas> From: "N. J. A. Sloane" <njas at research.att.com>
njas> To: franktaw at netscape.net, seqfan at ext.jussieu.fr
njas> Subject: Re: Partitions invariant under a permutation
njas> Cc: njas at research.att.com
njas> 
njas> ON THE OTHER HAND, I agree that your formula,
njas> > e^{Sum_{d|l} (e^{d*x}-1)/d}. 
njas> , is better!   So let's add your versions of 
njas> A036074, A036076, A036078, A036079 A036080, and A036082
njas> as new sequences - will you please send them in in the usual way?

For n=prime, these results match the "sorting numbers." The proposed
formula  for the general case n=4,6,8,9,... does not get OEIS "hits":

for n from 1 to 15 do
od:

E.g.f. exp[sum_{d|n} (exp(dn)-1)/d], n=1 A000110
1,1,2,5,15,52,203,877,4140,21147,115975,678570,4213597,27644437,190899322,1382958545,10480142147,82864869804,682076806159,5832742205057,51724158235372,

E.g.f. exp[sum_{d|n} (exp(dn)-1)/d], n=2 A002872
1,2,7,31,164,999,6841,51790,428131,3827967,36738144,376118747,4086419601,46910207114,566845074703,7186474088735,95318816501420,1319330556537631,19013488408858761,284724852032757686,4422344774431494155,

E.g.f. exp[sum_{d|n} (exp(dn)-1)/d], n=3 A002874
1,2,8,42,268,1994,16852,158778,1644732,18532810,225256740,2933174842,40687193548,598352302474,9290859275060,151779798262202,2600663778494172,46609915810749130,871645673599372868,16971639450858467002,343382806080459389676,

E.g.f. exp[sum_{d|n} (exp(dn)-1)/d], n=4
1,3,16,111,931,9066,99925,1224579,16466524,240481659,3783550243,63705870894,1141642250521,21673475232051,434151377912680,9144596750676663,201921333704487283,4661560653703827474,112247618536314408013,2813147392487637246483,73238904876868407843892,

E.g.f. exp[sum_{d|n} (exp(dn)-1)/d], n=5 A036075
1,2,10,70,602,6078,70402,917830,13253002,209350350,3584098770,66012131222,1300004931162,27232369503902,604103160535330,14136908333006822,347827448896896554,8971450949011952494,241945797101106717042,6806402312564710930742,199320131212209434124026,

E.g.f. exp[sum_{d|n} (exp(dn)-1)/d], n=6
1,4,28,258,2892,37778,560124,9256010,168182044,3325057826,70934634236,1621828212826,39517131361884,1021237022557682,27877344103738940,800976143703407210,24148078430008534428,761815206361252780098,25087729474993723079548,860535645960352227910650,30684369076328548340402908,

E.g.f. exp[sum_{d|n} (exp(dn)-1)/d], n=7 A036077
1,2,12,106,1144,14434,209736,3451290,63194936,1269555762,27700698344,651497885482,16414347638936,440651469115394,12546081858835528,377328994871025210,11946046637611280120,396918275031007485266,13803266030274159058856,501211773705339167558346,18961794630356792861173080,

E.g.f. exp[sum_{d|n} (exp(dn)-1)/d], n=8
1,4,31,329,4316,66543,1172077,23139068,504673027,12023659317,310112290044,8596075462411,254551364575929,8012189624924740,266901629110149847,9374566215360038977,346044223246303101068,13385712345895131535991,541214970909342384439077,22820383977428362514102620,1001398316281511650124170331,

E.g.f. exp[sum_{d|n} (exp(dn)-1)/d], n=9
1,3,22,235,3139,49376,891491,18160071,411124024,10212362573,275543711899,8012352809874,249516568317597,8277748828228015,291219448755079278,10821916875914147551,423314732431005599479,17377262163536437555816,746608679347375235630183,33493344279772838697144275,1565463366924934211448932472,

E.g.f. exp[sum_{d|n} (exp(dn)-1)/d], n=10
1,4,34,410,6170,109506,2227882,50987378,1293386410,35933906946,1083069266634,35146231883122,1220316225699786,45099576850647106,1766204747811044266,73012853049788936050,3175271515525778312810,144842213461353105363458,6911904715782802105599498,344243954984179241931745522,17855898233175714169836841610,

E.g.f. exp[sum_{d|n} (exp(dn)-1)/d], n=11 A036081
1,2,16,202,3044,52794,1055260,24081754,615896308,17347970202,531721375308,17595339114554,624882463734756,23691503493287738,954301756159098172,40665568780962213530,1826521141853468785364,86200513495391202853210,4262663346817813209581868,220325236503791932786240634,11876532766522864277257351364,

E.g.f. exp[sum_{d|n} (exp(dn)-1)/d], n=12
1,6,64,930,16780,356922,8681380,236739354,7135453180,235079261562,8389974421012,322019941283514,13211434169884204,576435716324437722,26631611751773999044,1297943388416061780186,66511484156498006072668,3573216982590418423591866,200735280710053376309792116,11764880374853891571677636730,717875316484862234010796972684,

E.g.f. exp[sum_{d|n} (exp(dn)-1)/d], n=13
1,2,18,262,4498,88174,1989162,51366438,1491069602,47749828830,1664928894170,62693869629142,2534737217687378,109469680507411214,5025930552213949450,244236790780300327302,12515419830686362586882,674102621202004022204990,38057622607352031273487418,2246533780909327553713483062,138344951637859243558826010546,

E.g.f. exp[sum_{d|n} (exp(dn)-1)/d], n=14
1,4,40,602,11384,253698,6495616,187940402,6055160152,214444348674,8261055193008,343357357970866,15298169514407880,726752780487465922,36642712015230282784,1952976182776961017138,109644342878030261599544,6464171341682259496350722,399107882296854625719568272,25743126812332530931279791154,1730916907343385903833629046696,

E.g.f. exp[sum_{d|n} (exp(dn)-1)/d], n=15
1,4,40,612,11976,276836,7336248,219610532,7331824360,269576919908,10796426089880,466904884431268,21658080409800264,1071807203930733668,56328256034260866296,3131089417758323092388,183430891419064587294120,11289874100345545058844516,728012324282559431644621912,49062570250294696515147333540,3447973479702517522741813507336,

-------
We may also drop the outer exponentiation and look for sequences
that relate to their E.g.f's. These turn out to be generally in the OEIS
for n<=12 (see below), and essentially we've discovered E.g.f.'s for these.
The relation to what's shown above is the exponential transform (and that
I'd suspect to be a "well known" property for the sum of the n'th powers
of the divisors of k.).

for n from 1 to 15 do
od:

E.g.f. sum_{d|n} (exp(dn)-1)/d, n=1 A000012
0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,

E.g.f. sum_{d|n} (exp(dn)-1)/d, n=2 A000051
0,2,3,5,9,17,33,65,129,257,513,1025,2049,4097,8193,16385,32769,65537,131073,262145,524289,

E.g.f. sum_{d|n} (exp(dn)-1)/d, n=3 A034472
0,2,4,10,28,82,244,730,2188,6562,19684,59050,177148,531442,1594324,4782970,14348908,43046722,129140164,387420490,1162261468,

E.g.f. sum_{d|n} (exp(dn)-1)/d, n=4 A001576
0,3,7,21,73,273,1057,4161,16513,65793,262657,1049601,4196353,16781313,67117057,268451841,1073774593,4295032833,17180000257,68719738881,274878431233,

E.g.f. sum_{d|n} (exp(dn)-1)/d, n=5  A034474
0,2,6,26,126,626,3126,15626,78126,390626,1953126,9765626,48828126,244140626,1220703126,6103515626,30517578126,152587890626,762939453126,3814697265626,19073486328126,

E.g.f. sum_{d|n} (exp(dn)-1)/d, n=6 A034488
0,4,12,50,252,1394,8052,47450,282252,1686434,10097892,60526250,362976252,2177317874,13062296532,78368963450,470199366252,2821153019714,16926788715972,101560344351050,609360902796252,

E.g.f. sum_{d|n} (exp(dn)-1)/d, n=7 A034491
0,2,8,50,344,2402,16808,117650,823544,5764802,40353608,282475250,1977326744,13841287202,96889010408,678223072850,4747561509944,33232930569602,232630513987208,1628413597910450,11398895185373144,

E.g.f. sum_{d|n} (exp(dn)-1)/d, n=8 A034496
0,4,15,85,585,4369,33825,266305,2113665,16843009,134480385,1074791425,8594130945,68736258049,549822930945,4398314962945,35185445863425,281479271743489,2251816993685505,18014467229220865,144115462954287105,

E.g.f. sum_{d|n} (exp(dn)-1)/d, n=9 A034513
0,3,13,91,757,6643,59293,532171,4785157,43053283,387440173,3486843451,31381236757,282430067923,2541867422653,22876797237931,205891146443557,1853020231898563,16677181828806733,150094635684419611,1350851718835253557,

E.g.f. sum_{d|n} (exp(dn)-1)/d, n=10 A034517
0,4,18,130,1134,10642,103158,1015690,10078254,100390882,1001953638,10009766650,100048830174,1000244144722,10001220711318,100006103532010,1000030517610894,10000152587956162,100000762939584198,1000003814697527770,10000019073486852414,

E.g.f. sum_{d|n} (exp(dn)-1)/d, n=11 A034524
0,2,12,122,1332,14642,161052,1771562,19487172,214358882,2357947692,25937424602,285311670612,3138428376722,34522712143932,379749833583242,4177248169415652,45949729863572162,505447028499293772,5559917313492231482,61159090448414546292,

E.g.f. sum_{d|n} (exp(dn)-1)/d, n=12 A034660
0,6,28,210,2044,22386,257908,3037530,36130444,431733666,5170140388,61978939050,743375541244,8918294543346,107006334784468,1283997101947770,15407492847694444,184887084343023426,2218628050709022148,26623434909949071690,319480609006403630044,

E.g.f. sum_{d|n} (exp(dn)-1)/d, n=13 
0,2,14,170,2198,28562,371294,4826810,62748518,815730722,10604499374,137858491850,1792160394038,23298085122482,302875106592254,3937376385699290,51185893014090758,665416609183179842,8650415919381337934,112455406951957393130,1461920290375446110678,

E.g.f. sum_{d|n} (exp(dn)-1)/d, n=14
0,4,24,250,3096,40834,554664,7647250,106237176,1481554114,20701400904,289537131250,4051542498456,56707753666594,793811662272744,11112685048647250,155572843119354936,2177986570740006274,30491579359845314184,426880482624234915250,5976315357844100294616,

E.g.f. sum_{d|n} (exp(dn)-1)/d, n=15
0,4,24,260,3528,51332,762744,11406980,170939688,2563287812,38445332184,576660215300,8649804864648,129746582562692,1946196290656824,29192932133689220,437893920912786408,6568408508343827972,98526126098761952664,1477891883850485076740,22168378219605654448968,