Real-rooted atlas

A034870 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(4 + 32\,x + 88\,x^{2} + 96\,x^{3} + 36\,x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
14243664
216192864172812964096
3641152864034560777609331246656262144
42566144645123870721451520348364852254724478976167961616777216
510243072041472033177601741824062705664156764160268738560302330880201553920604661761073741824
64096147456243302424330240164229120788299776275904921670946979841330255872017736744960159630704648707129344217678233668719476736
7163846881281341849616102195213284311047970586624358676398081229747650563228087582726456175165449684262748161056465027072792348770304365699432448783641640964398046511104
86553631457287077888099090432096613171206956148326438258815795216396635340805533864427520147569718067203098964079411250710321299456633879016243205851190919168037614798766080150459195064322821109907456281474976710656
9262144141557763609722885775556608649750118405457900994563547635646464182449833246727526055621427225086852071424067734500592844814778436492984322586226386272256358092884253081638367094741401603069367579312128172651926336307260935974001049610155995666841618014398509481984
101048576629145601793064960322751692804115084083203950480719872296286053990401777716323942408666367079219203466546831687680114396045445693443119892148518912070197573341675520129595520015400960194393280023101440233271936027721728218692440025989120154371134135992320771855670679961602437438960041984036561584400629761152921504606846976
11419430427682406487199580161743991603202485188034560268400307732482281402615726081564390365069312879969580351488041065247083069440160154463623970816524141880951177216144139017261573734433262850137286246406414978240762347520102639651852197560321347145430560092979214263892794165690368118865773284714086407507311996929310720337829039861818982496522582817662566413162170384226713673786976294838206464
121677721612079595524167460454491684129996814440250474496173283005693952164618855409254412699168845856768809572013923368964317717407591301121942972833416085504741862354577050828824110526523754151936667676119119345745921573807995067029258243147615990134058516485311601983351223746567498732211789962936328748520913754956759048288072444609959034886216054333457469276163552031047689982443521453103610418629181443790705070657293516847383813383216168964722366482869645213696
136710886452344913921962934272004711042252800812654788608001072704320962560112633953701068809654338888663040068787164581724160041272298749034496002104887236200759296091849624852396769280344436093196487884800111279353186557624320030999248387683909632007439819613044138311680153446279519035352678402707875520924153282560040618132813862299238400513071151332997464064005387247088996473372672046176403619969771765760314839115590702989312001642638863951493857280061598957398181019648001478374977556344471552170581728179578208256302231454903657293676544
142684354562254857830491321742131223743652954112445193492889600641078629761024073724042422517760695112399983738880547401014987194368036493400999146291200208012385695133859840102115171159429349376043398947742757473484801602422685886429790208051506443474920957542400144218041729778681118720351531476716335535226880744419597752239956951040136476926254577325441024021548988355985893490688002909113428058095621242880332470106063782356713472031735782851542861322649602483669962294658712207360155229372643416169512960074510098868839761366220825791957300752225088307257315460668338277974016614094221446481549721619342813113834066795298816
Technical details
Node id
n-db2d0b2ce9624a44 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-12929af9fbcc0757

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:      4,       24,        36;
  2:     16,      192,       864,       1728,        1296;
  3:     64,     1152,      8640,      34560,       77760,        93312,         46656;
  4:    256,     6144,     64512,     387072,     1451520,      3483648,       5225472,        4478976,        1679616;
  5:   1024,    30720,    414720,    3317760,    17418240,     62705664,     156764160,      268738560,      302330880,       201553920,        60466176;
  6:   4096,   147456,   2433024,   24330240,   164229120,    788299776,    2759049216,     7094697984,    13302558720,     17736744960,     15963070464,       8707129344,       2176782336;
  7:  16384,   688128,  13418496,  161021952,  1328431104,   7970586624,   35867639808,   122974765056,   322808758272,    645617516544,    968426274816,    1056465027072,     792348770304,     365699432448,      78364164096;
  8:  65536,  3145728,  70778880,  990904320,  9661317120,  69561483264,  382588157952,  1639663534080,  5533864427520,  14756971806720,  30989640794112,   50710321299456,   63387901624320,   58511909191680,   37614798766080,   15045919506432,    2821109907456;
  9: 262144, 14155776, 360972288, 5775556608, 64975011840, 545790099456, 3547635646464, 18244983324672, 75260556214272, 250868520714240, 677345005928448, 1477843649298432, 2586226386272256, 3580928842530816, 3836709474140160, 3069367579312128, 1726519263363072, 609359740010496, 101559956668416;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (4+32*x+88*x^2+96*x^3+36*x^4)*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[Together[((4 + 32*x + 88*x^2 + 96*x^3 + 36*x^4)*P[n - 1, x])/(1 + 2*x + x^2)]];
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] = ((4 + 32*x + 88*x^2 + 96*x^3 + 36*x^4)*v[i - 1])/(1 + 2*x + x^2)); 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.