Real-rooted atlas

A034870 → basis A013611 → x ↦ x+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(36 + 48\,x + 16\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
1364816100
2129634563456153625610000
3466561866243110402764801382403686440961000000
41679616895795220901888278691842322432012386304412876878643265536100000000
5604661764031078401209323520214990848025082265602006581248111476736042467328010616832015728640104857610000000000
62176782336174142586886385228185614189395968021284093952022703033548817657914982410090237132842042654720124570828802491416576301989888167772161000000000000
7783641640967313988648963169395081216845172021657615494820397056206597605294082065976052940815740769927168918211579084840809403514881360313450496329772957696549621596165637144576268435456100000000000000
8282110990745630091839012864150459195064320468095273533440101420642598912016227302815825921983337010823168188889239126016014166692934451208395077294489603917702737428481424619177246723957275492352081174881894401159641169920103079215104429496729610000000000000000
910155995666841612187194800209926906077053452288245549406344970246138735158624256011458972296098611216551848872142438418916398711019929617340032151768268812844468260569088077066809563414528373657258489282561453111560791654444711124947435521064550593986560189253438930944236566798663681855425871872687194767361000000000000000000
103656158440062976487487792008396803087422682719846401234969073087938560349907904041582592074647019528870952961244116992147849216016588226561971322880179705787754689331201597384780041682944011714155053639008256709948791129636864035497439556481843201456305212573614080485435070857871360129449352228765696269686150476595204230370987868160470041220874240329853488332801099511627776100000000000000000000
111316217038422671361930451656353251328135131615944727592966005849597543448576019018523725554253824045644456941330209177686217307555845950668813137875437081287720961642234429635160965120170305792702905581568014759835367585150402561073442572188010938368655992683003784462336336406504104504852480144174216044787793920512619434825912156161495140018242243788835179765135111618566514771321316966409143538696585216091435386965852165805421394657281759218604441610000000000000000000000
1247383813383216168967581410141314587033658124144416745167257628416248381519859548169945686933531950841856265218318227518689116165599053384803172325785695983772310911525584896135977010773791327911936161157938694863796043776161157938694863796043776136740069195642008764416987567166412970063298566077336408695200389529631833666902689144897536141482964011951755100165305611150448190816256166450545896413829529643153845232403585433690850200489270706176151417000815451176961922755565910491136174795960537317376101330991615836162814749767106561000000000000000000000000
1317058172817957820825629567499551126889431042463958295927240785920013141110911611950858240050374258494512478289920014776449158390326965043203447838136957742958510080656731073706236754001920010398242000348748605030400138643226671316648067072001571289902274922011426816015236750567514395262320640126972921395953293852672009116004613042800584294400564324095093125750456320030097285071633373357670401379458899116529612226560540964274163344945971200180321424721114981990400506165402725936791552001181052606360519180288022496240121152746291203408521230477688832003951908673017610240032932572275146752001756403854674493444503599627370496100000000000000000000000000
146140942214464815497216114630921336676555948032103167829203008900353228859608079095071809092976642483669962294658712207360079477438793429078790635520203109010249874312464957440425561735761641416593244160744733037582872479038177280110330820382647774672322560013975237248468718125160857601524571336196587431835729920143987292863011035228930048011814341978503469557245542408438815698931049683746816005250818657112653136553574402844193439269353782299852801338443971420872368141107205452919883566517055389696019133052223040410720665600573991566691212321619968014577563598506979596697603092210460289359308390405377757322242364014592074691073920032833536007967047884803502243846128498372925770956830264189495929733127205759403792793610000000000000000000000000000
Technical details
Node id
n-fd18f6eb36bb6ac9 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-6b852c5cd7d4c499

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:               1;
  1:              36,               48,               16;
  2:            1296,             3456,             3456,              1536,               256;
  3:           46656,           186624,           311040,            276480,            138240,              36864,               4096;
  4:         1679616,          8957952,         20901888,          27869184,          23224320,           12386304,            4128768,             786432,              65536;
  5:        60466176,        403107840,       1209323520,        2149908480,        2508226560,         2006581248,         1114767360,          424673280,          106168320,           15728640,           1048576;
  6:      2176782336,      17414258688,      63852281856,      141893959680,      212840939520,       227030335488,       176579149824,       100902371328,        42042654720,        12457082880,        2491416576,         301989888,          16777216;
  7:     78364164096,     731398864896,    3169395081216,     8451720216576,    15494820397056,     20659760529408,     20659760529408,     15740769927168,      9182115790848,      4080940351488,     1360313450496,      329772957696,       54962159616,       5637144576,        268435456;
  8:   2821109907456,   30091839012864,  150459195064320,   468095273533440,  1014206425989120,   1622730281582592,   1983337010823168,   1888892391260160,   1416669293445120,    839507729448960,   391770273742848,   142461917724672,    39572754923520,    8117488189440,    1159641169920,    103079215104,     4294967296;
  9: 101559956668416, 1218719480020992, 6906077053452288, 24554940634497024, 61387351586242560, 114589722960986112, 165518488721424384, 189163987110199296, 173400321517682688, 128444682605690880, 77066809563414528, 37365725848928256, 14531115607916544, 4471112494743552, 1064550593986560, 189253438930944, 23656679866368, 1855425871872, 68719476736;
%F Formula
The row polynomials satisfy P(n,x) = (36+48*x+16*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(36 + 48*x + 16*x^2)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (36 + 48*x + 16*x^2)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.