Real-rooted atlas

A038226 → basis A013609 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(27 + 16\,x + 16\,n\right)\,P_{n}(x) = \left(473 + 864\,x + 256\,x^{2} + 176\,n + 256\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 473 + 864\,x + 256\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
04738642561593
171392270420736409654675
299825456896726528442368655361791153
31332331798600018275840193740808847360104857656864187
417232457127903776383328000584826880464977920169869312167772161764915561
521790200319300351847162611456143109120001637515264010415505408317089382426843545653851447971
62708716769278914563841235222517763056047554564579491840004192039075842221974487045798205849642949672961621354746465
73322562655539005521473620081848596485929068085248110017711964161318893649920010060893782016457091894476810436770529286871947673648295450738251
8403209055161531610024880031204417178880107103192514560237162723409920352056678285312351704973312000229963286446080914183788953601855425871872010995116277761425986729692569
9485029840038770964793708336467816821894400183065914116096047125404706406408348127864029184103269958963691528842867900416000505919230181376017877371872870403265549534494721759218604441641795899815213747
1057918269134033931257292874304681262019600025629940276601241600878716387757260801809615540726005762671400916489338882832547445861253122122288296099840001079294357720268803432455399591116856998682783907842814749767106561217434264075659537
116873158145018991204699997987886496850758458927616472341666922684416155689438326456320036554601730702049286273333874516819968793787700899689267273646233592392581124904844062097408000224493226405815910464904611192268390498797716825440256450359962737049635272239294794793435
12811283735382378515395874244842537613492639977464327687231523298266595328264511333476703272966974886837025243136013647051312795431731220074668398453823897622226055625191299481618329729249662153523210986850699098193920045714984286275239936121155274225567662081702360659146047488720575940379279361017191343407763043017
1395317208229393491194708096491770840018475049093811045120107941119819714621440433891397895995719680126965440068817571020827899547348100972544004678989021529862307840602240051953614716928059269481667176798617604399135019918916845568239713106162142412800091429968572550479872022367127549335568384029183325585360814080115292150460684697629214870098113799908803
141115325261233978817244012053067247336962492263635094666752001576537522671875850240690823166846173577216022215239572274980847616541719210960288303022081020326303016264138752001497276486889555938508801713038370001392972595201517298730679726044610561023798695544766102241285113879598125704806400018004670734286863728640408998903759278964736049806208999015789363218446744073709551616836070495953885936715969
Technical details
Node id
n-f74afc0d030d86d2 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-372f8f7088769244

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:           473,            864,             256;
  1:          7139,          22704,           20736,             4096;
  2:         99825,         456896,          726528,           442368,            65536;
  3:       1332331,        7986000,        18275840,         19374080,          8847360,          1048576;
  4:      17232457,      127903776,       383328000,        584826880,        464977920,        169869312,          16777216;
  5:     217902003,     1930035184,      7162611456,      14310912000,      16375152640,      10415505408,        3170893824,        268435456;
  6:    2708716769,    27891456384,    123522251776,     305604755456,     457949184000,     419203907584,      222197448704,      57982058496,       4294967296;
  7:   33225626555,   390055214736,   2008184859648,    5929068085248,   11001771196416,   13188936499200,    10060893782016,    4570918944768,    1043677052928,      68719476736;
  8:  403209055161,  5316100248800,  31204417178880,  107103192514560,  237162723409920,  352056678285312,   351704973312000,  229963286446080,   91418378895360,   18554258718720,   1099511627776;
  9: 4850298400387, 70964793708336, 467816821894400, 1830659141160960, 4712540470640640, 8348127864029184, 10326995896369152, 8842867900416000, 5059192301813760, 1787737187287040, 326554953449472, 17592186044416;
%F Formula
The row polynomials satisfy (16*n+27+16*x)*P(n,x) = (176*n+473+(256*n+864)*x+256*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 473 + 864*x + 256*x^2.
%t Mathematica
P[0, x_] := 473 + 864*x + 256*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((176*n + 473 + (256*n + 864)*x + 256*x^2)*P[n - 1, x])/(16*n + 27 + 16*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 473 + 864*x + 256*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((176*n + 473 + (256*n + 864)*x + 256*x^2)*v[i - 1])/(16*n + 27 + 16*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.