Real-rooted atlas

A038320 → x ↦ x+1 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(23 + 6\,x + 6\,n\right)\,P_{n}(x) = \left(493 + 276\,x + 36\,x^{2} + 102\,n + 36\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 493 + 276\,x + 36\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
049327636805
1101158874248421621689
2201433242760106488198721296571849
33925487604299036414001064880149040777614831573
475252421141317532108773820436968009583920107308846656379744237
514241165713160601682296766817215228334804588164008050492875116162799369622332785
626672013745683575954087585444036847482690752182740017604404637440644039424515082241679616241742606737
7495278778311144028874223018456550760161365379926624641016325152197359219008396417369604968304128347680512100776966029945866637
89131266490269297167266986604320866226690036913101520320204806988993607692195901824197359219008033978631680037262280960231787008060466176149495670901429
916732749373726760266358835775498065198105578095059056987180060906617508528027034522547155284614154920064186081549350402803237113600273256727040152979425283627970563686959498014761
1030501987713793371204757954908322421695889180879144235356475453387201711063025769240087705529212280323244142705658624870317022034944167473394415360224258969088001967448434688100131987456217678233690516927001823065
1155349112536827295237915504167588286469855602414245736564093118702857744458945127134105040266925832020001440114017187975964416361490187201960968485590964840704145143608493312017492199588864013950997991424650857918464130606940162213377067634052421
121000362381323499637464932545309349278099924511750387080121315595686759888060811845955492760012624771027813585296467237369616482800201601368206255711572992379564696562059008791988490051799041219206311343820813357679686041609765698593996842055434731527836416409653931890727202009981
131801642744185589940390032614319114967330209219645389207175100299773535251161240360296009029520974813680213227198869680227232115654172037794470080480466497636002592001539232037675519616037956469656205900807127896410466191369975324365540352010018259764531200676086825738240270356366131204701849845761309990504672878728417
143234538048206091846091729577034418166342688432156548731751843184066950286524546296032007194564846027869768640568337336680271642265634116351819148836357121586114359375467018240576559797163203110400164184750685388759040364382108699576647686220709594588676096798025949243228160739809951842304004636023947919360173028074323968282110990745631729593020262558227233
Technical details
Node id
n-2d66ee0996e2bc65 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-a784d59e7b626275

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:             493,             276,              36;
  1:           10115,            8874,            2484,             216;
  2:          201433,          242760,          106488,           19872,            1296;
  3:         3925487,         6042990,         3641400,         1064880,          149040,            7776;
  4:        75252421,       141317532,       108773820,        43696800,         9583920,         1073088,          46656;
  5:      1424116571,      3160601682,      2967668172,      1522833480,       458816400,        80504928,        7511616,         279936;
  6:     26672013745,     68357595408,     75854440368,     47482690752,     18274001760,      4404637440,      644039424,       51508224,       1679616;
  7:    495278778311,   1440288742230,   1845655076016,   1365379926624,    641016325152,    197359219008,    39641736960,     4968304128,     347680512,     10077696;
  8:   9131266490269,  29716726698660,  43208662266900,  36913101520320,  20480698899360,   7692195901824,  1973592190080,   339786316800,   37262280960,   2317870080,    60466176;
  9: 167327493737267, 602663588357754, 980651981055780, 950590569871800, 609066175085280, 270345225471552, 84614154920064, 18608154935040, 2803237113600, 273256727040, 15297942528, 362797056;
%F Formula
The row polynomials satisfy (6*n+23+6*x)*P(n,x) = (102*n+493+(36*n+276)*x+36*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 493 + 276*x + 36*x^2.
%t Mathematica
P[0, x_] := 493 + 276*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((102*n + 493 + (36*n + 276)*x + 36*x^2)*P[n - 1, x])/(6*n + 23 + 6*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 493 + 276*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((102*n + 493 + (36*n + 276)*x + 36*x^2)*v[i - 1])/(6*n + 23 + 6*x)); 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

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.