Real-rooted atlas

A038326 → 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

\(\left(11 + 12\,x + 12\,n\right)\,P_{n}(x) = \left(529 + 552\,x + 144\,x^{2} + 276\,n + 144\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1351247
210818401442065
3312113891615120172886975
48638571498128933984241920207363558625
5232268035183142044943840186796803628800248832142559375
66114525851672329816186593112010786521603362342405225472029859845619840625
71583984012351362017140702378522725224607136022651695360564873523273156608035831808218753609375
8405174228193152062465180824653768227202247611272704125390571264043491255091290379763712100329062404299816968428447890625
9102587390718114375881664484482113731197632887535656179206068550436300827084363393024782842591641613944306401281354442342405159780352322018180859375
10257564830589209123104868861732026255289986906403284549247905280266260696853760014564521047121925416872678604801342015871385602091645960192018059231232006191736422412216423994140625
1164211092381155953399855763777558881249213448743120115523275942388160108390125180874240702928239693926403204194630366822410214674193940480221432618778624030677474082816023838185226240743008370688460677408037109375
1215911922957162580992463973028864568024478961499198423363899962245539669760415883793392597376031216356052091781121687027775265423360659148609675460608183864135490928640354292190045798404417556267925504312063515688960891610044825617282299169775390625
13392271777333163745924822599813173626204721218989625143630401272905997958318014721520985275760471206401297557435384903813128116252573543863091237596618991629434880128533978886714818563186978348509429760552695816471445504626489797996707844056825703956480106993205379072645431117038818359375
149627094512986527015365901658591971509311220850983843065846011364038826341900804330240534620519142493566182451105105265551832535043633160819077730676736194790061765052714188878952899882421813248023993009392186766131253541236254958419968844117246974571315287708571719539097652426670635745280128391846454886424009361709742431640625
Technical details
Node id
n-e6a3dd03637ddeee · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-10435174f4c6490a

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:             35,             12;
  2:           1081,            840,            144;
  3:          31211,          38916,          15120,           1728;
  4:         863857,        1498128,         933984,         241920,          20736;
  5:       23226803,       51831420,       44943840,       18679680,        3628800,         248832;
  6:      611452585,     1672329816,     1865931120,     1078652160,      336234240,       52254720,       2985984;
  7:    15839840123,    51362017140,    70237852272,    52246071360,    22651695360,     5648735232,     731566080,      35831808;
  8:   405174228193,  1520624651808,  2465376822720,  2247611272704,  1253905712640,   434912550912,   90379763712,   10032906240,    429981696;
  9: 10258739071811, 43758816644844, 82113731197632, 88753565617920, 60685504363008, 27084363393024, 7828425916416, 1394430640128, 135444234240, 5159780352;
%F Formula
The row polynomials satisfy (12*n+11+12*x)*P(n,x) = (276*n+529+(144*n+552)*x+144*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[Together[((276*n + 529 + (144*n + 552)*x + 144*x^2)*P[n - 1, x])/(12*n + 11 + 12*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] = ((276*n + 529 + (144*n + 552)*x + 144*x^2)*v[i - 1])/(12*n + 11 + 12*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. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.