Real-rooted atlas

A103327 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(-1 - 2\,x - x^{2} + 2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(-4 - 21\,x - 38\,x^{2} - 29\,x^{3} - 8\,x^{4} + 8\,n + 34\,n x + 52\,n x^{2} + 34\,n x^{3} + 8\,n x^{4}\right)\,P_{n-1}(x) + \left(-4\,x - 20\,x^{2} - 36\,x^{3} - 28\,x^{4} - 8\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1459
216442989
364272376169881
42561472312028889858721
510247424212483000020892574186329
640963584012928024601626044814542833461854569
7163841679367311361752320249664021144009855201950258459361
8655367700483928064113602562037120023191424163680326547792113668983739041
926214434734082031616068841472148915200213240576202124032122284960428479886625109828931049
101048576154664961020395523964928001004781568173509324820675683841678692608888733680277040572386139658205571449
1141943046815744050069504021947351046377766912129002209281853191782418906940160134249663366318279408177385130422505868181226783441
1216777216297795584241067622411767644160385774387208947099238415052421529618508417433616507393792010414422304044116347984112663019761311738121804062262961
13671088641291845632114252840966146647654422441492480058715386675211325450485761630581768192175226887987213881135795207879057638403034450164007107091303676453700457959395846169
1426843545655700357125343543296031411981516812640478822403683337371648801452595609613228126371840166420419133441588098342297611315230594560583707042560020608374025604457406564204456048214978789896198729
Technical details
Node id
n-3ac8a6274ff270bb · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-9247102c6b6e4eea

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,       5;
  2:     16,      44,       29;
  3:     64,     272,      376,      169;
  4:    256,    1472,     3120,     2888,       985;
  5:   1024,    7424,    21248,    30000,     20892,      5741;
  6:   4096,   35840,   129280,   246016,    260448,    145428,     33461;
  7:  16384,  167936,   731136,  1752320,   2496640,   2114400,    985520,    195025;
  8:  65536,  770048,  3928064, 11360256,  20371200,  23191424,  16368032,   6547792,  1136689;
  9: 262144, 3473408, 20316160, 68841472, 148915200, 213240576, 202124032, 122284960, 42847988, 6625109;
%F Formula
The row polynomials satisfy (2*n-1+(4*n-2)*x+(2*n-1)*x^2)*P(n,x) = (8*n-4+(34*n-21)*x+(52*n-38)*x^2+(34*n-29)*x^3+(8*n-8)*x^4)*P(n-1,x) - (4*x+20*x^2+36*x^3+28*x^4+8*x^5)*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[((8*n - 4 + (34*n - 21)*x + (52*n - 38)*x^2 + (34*n - 29)*x^3 + (8*n - 8)*x^4)*P[n - 1, x] - (4*x + 20*x^2 + 36*x^3 + 28*x^4 + 8*x^5)*D[P[n - 1, x], x])/(2*n - 1 + (4*n - 2)*x + (2*n - 1)*x^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] = ((8*n - 4 + (34*n - 21)*x + (52*n - 38)*x^2 + (34*n - 29)*x^3 + (8*n - 8)*x^4)*v[i - 1] - (4*x + 20*x^2 + 36*x^3 + 28*x^4 + 8*x^5)*deriv(v[i - 1], x))/(2*n - 1 + (4*n - 2)*x + (2*n - 1)*x^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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [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
  3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023