Real-rooted atlas

A196347 → basis A013611 → basis A013609

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(6\,n + 8\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
16814
272192128392
3129651846912307216464
43110416588833177629491298304921984
59331206220800165888002211840014745600393216064538880
633592320268738560895795200159252480015925248008493465601887436805421265920
7141087744013168189440526727577601170505728001560674304001248539443205549064192010569646080531284060160
86772211712072236924928033710564966408989483991040149824733184001598130487296010654203248640405874409472067645734912059503814737920
9365699432448043883931893760234047636766720728148203274240145629640654848019417285420646401725980926279680986274815016960328758271672320487049291366407497480656977920
1021941965946880029255954595840001755357275750400062412703137792000145629640654848000233007425047756800258897138941952000197254963003392000986274815016960002922295748198400038963943309312001049647291976908800
1114481697524940800212398230365798400141598820243865600056639528097546240001510387415934566400028193898430778572800375918645743714304003580177578511564800023867850523410432000106079335659601920002828782284256051200342882701121945600161645682964443955200
1210426822217957376001668291554873180160012234138069069987840054373946973644390400016312184092093317120003479932606313240985600541322849870948597760061865468556679839744005155455713056653312000305508486699653529600012220339467986141184002962506537693609984003291673930770677760027156474738026584473600
138132921330006753280014097063638678372352001127765091094269788160055135182231275411865600183783940770918039552000441081457850203294924800784144813955916968755200104552641860788929167360010455264186078892916736007744640137836216975360004130474740179315720192001501990814610660261888003337757365801467248640034233408880015048704004942478402320838374195200
14683165391720567275520012752420645450589143040011052097892723843923968005894452209452716759449600216129914346599614513152005763464382575989720350720011526928765151979440701440017564843832612540100116480020492317804714630116802560018215393604190782326046720012143595736127188217364480058878039932737882266009600196260133109126274220032004025848884289769727590400383414179456168545484800968725766854884321342259200
Technical details
Node id
n-69b8eb7cf815c14f · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-c91d5d506c107735

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, 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:             6,              8;
  2:            72,            192,             128;
  3:          1296,           5184,            6912,            3072;
  4:         31104,         165888,          331776,          294912,            98304;
  5:        933120,        6220800,        16588800,        22118400,         14745600,          3932160;
  6:      33592320,      268738560,       895795200,      1592524800,       1592524800,        849346560,        188743680;
  7:    1410877440,    13168189440,     52672757760,    117050572800,     156067430400,     124853944320,      55490641920,     10569646080;
  8:   67722117120,   722369249280,   3371056496640,   8989483991040,   14982473318400,   15981304872960,   10654203248640,   4058744094720,    676457349120;
  9: 3656994324480, 43883931893760, 234047636766720, 728148203274240, 1456296406548480, 1941728542064640, 1725980926279680, 986274815016960, 328758271672320, 48704929136640;
%F Formula
The row polynomials satisfy P(n,x) = (6*n+8*n*x)*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[(6*n + 8*n*x)*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] = (6*n + 8*n*x)*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. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.