Real-rooted atlas

A114687 → basis A013609 → reverse

nonnegative coefficients

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

Recurrence

\(\left(2 + n\right)\,P_{n}(x) = \left(8 + 8\,x + x^{2} + 4\,n + n x - n x^{2}\right)\,P_{n-1}(x) + \left(8\,x + 6\,x^{2} + x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1437
216281155
36419216845469
4256115216649681974237
5102464001344012640550090339907
64096337929600012896089520310924279387739
71638417203263078411289601106560607488175504207933858505
86553685196838993928888320114903048834560400691299073610304939130777
92621444128768230031366468403210554163210625126467081728259032965597172518859402972031
1010485761966080013074432044285952088424448011162296329176832004909459201649863203165582026467234202705311
114194304922746887208960002887188480689356800010585817088108583157767531814400349349088010389667201792515121364886944299426717
121677721642781900838755368961808007168050704220160926041374721150214799369915099955259421415680243155616006484376448101625890471039373471189693925
1367108864196293427220392706048109460848640355410378752758625533952111596647219211606340894728616426424324542651210241662529440004017960835257685275963726935195051001609115
1426843545689254789121053441392646438969671682392094408704588445777920010077379952640123708213657601104744216576071957023057923384772495104112018633708824750069489632780930532196880151954513542257795
Technical details
Node id
n-c6340f52de30a0f5 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-f60b2631b48d7d2e

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:      4,       3;
  2:     16,      28,       11;
  3:     64,     192,      168,       45;
  4:    256,    1152,     1664,      968,       197;
  5:   1024,    6400,    13440,    12640,      5500,       903;
  6:   4096,   33792,    96000,   128960,     89520,     31092,     4279;
  7:  16384,  172032,   630784,  1128960,   1106560,    607488,   175504,    20793;
  8:  65536,  851968,  3899392,  8888320,  11490304,   8834560,  4006912,   990736,  103049;
  9: 262144, 4128768, 23003136, 64684032, 105541632, 106251264, 67081728, 25903296, 5597172, 518859;
%F Formula
The row polynomials satisfy (n+2)*P(n,x) = (4*n+8+(n+8)*x-(n-1)*x^2)*P(n-1,x) + (8*x+6*x^2+x^3)*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[((4*n + 8 + (n + 8)*x - (n - 1)*x^2)*P[n - 1, x] + (8*x + 6*x^2 + x^3)*D[P[n - 1, x], x])/(n + 2)]];
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] = ((4*n + 8 + (n + 8)*x - (n - 1)*x^2)*v[i - 1] + (8*x + 6*x^2 + x^3)*deriv(v[i - 1], x))/(n + 2)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023