Real-rooted atlas

A178822 → basis A013610 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(25 + 15\,x + 5\,n + 3\,n x\right)\,P_{n-1}(x) + \left(15\,x + 3\,n x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1301848
25258191891533
370002167212096151242280
478750437346445662129276102061101240
5787500747003612428640629370011226606123628163772
67218750113891778291729438229895820696049208419950336798721097454
761875000159906938460869128327017822504321080760064665216056582064173210418681453648
8502734375210895426171165990301191893291620221229104446573666888225052671839223490189568444007492724442925
9391015625026479373520620935568253604667297224590413383451481017143917450663595695863763869606941220096736663940536613278098936102
10293261718753195120525597357352594268671074342474305101264156176383107003400766862520119913582769312720984576026155311682510934989065177324147366363367720350
11213281250000373075530478565857064949415682343687625174896359471744311680025889847364180809787055060502722060613770735522467879261848016502967280805674372704077377809610361770623924720
12151074218750042372537985486892858166466261564897239050280376896533450828722908886321166650920004289267286858153611749425789589488188964791597181617969832030150098273042267402828158940883288556908300570569068409276
13104589843750004699848829017384143190220503395520987882808124950920247567269544158564028554129495526274641734130835962132608601148892648560256124681938828855408158113246120109281222685458394400557015769077041361462559912136601594648943921735645532608
14709716796875005106951315142981221567605334616896121935276210928074924061139400049339597440875835903192977727006575526524571083679628272357901302166080735465965540480460011942968483819573201224522380512070647850686403988792302540658422184635880423049255616363532272990845026544903164
Technical details
Node id
n-4d910af6db5dc7ca · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-c377b8329dc9dd7c

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          1;
  1:         30,           18;
  2:        525,          819,           189;
  3:       7000,        21672,         12096,          1512;
  4:      78750,       437346,        445662,        129276,         10206;
  5:     787500,      7470036,      12428640,       6293700,       1122660,         61236;
  6:    7218750,    113891778,     291729438,     229895820,      69604920,       8419950,       336798;
  7:   61875000,   1599069384,    6086912832,    7017822504,    3210807600,     646652160,     56582064,     1732104;
  8:  502734375,  21089542617,  116599030119,  189329162022,  122910444657,   36668882250,   5267183922,   349018956,    8444007;
  9: 3910156250, 264793735206, 2093556825360, 4667297224590, 4133834514810, 1714391745066, 359569586376, 38696069412, 2009673666, 39405366;
%F Formula
The row polynomials satisfy n*P(n,x) = (5*n+25+(3*n+15)*x)*P(n-1,x) + (3*n+15)*x*d/dx 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[((5*n + 25 + (3*n + 15)*x)*P[n - 1, x] + (3*n + 15)*x*D[P[n - 1, x], x])/(n)]];
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] = ((5*n + 25 + (3*n + 15)*x)*v[i - 1] + (3*n + 15)*x*deriv(v[i - 1], x))/(n)); 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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.