Real-rooted atlas

A033282 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, −1/2)coefficients of one signHadamard kernel

Recurrence

\(\left(2 + 4\,x + 2\,x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(10 + 44\,x + 70\,x^{2} + 48\,x^{3} + 12\,x^{4} - n - 11\,n x - 31\,n x^{2} - 33\,n x^{3} - 12\,n x^{4}\right)\,P_{n-1}(x) + \left(4 + 26\,x + 66\,x^{2} + 82\,x^{3} + 50\,x^{4} + 12\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1358
211373179
345228384215872
4197133433803798159710306
5903765525910437703690512425127568
6427943575184605416465527685356055999551632619
7207932472261258068355187060087306090980342573082467521428072
81030491401100832434428227640597537648085868068306728329353726939769286850446
95188597940169539459642135696645429660049192972601036556976750574404316711503593346053901415408
1026467234502050534427059515585598504625906040940581776413268441760128225835308124423495304762114551397296753759264374
1113648869255462240217142904011064440805375524143508913789469215100079610018255378537015435678109586935480060293528276364501041935748896002192
12710393731450880226135702440287686059667029360899408579692970064415759664805282287872520164241989738556518186749337909218805038522829893115543978403851710529556628996
1337269351982478995958418043787452464920937022277517580896805655598885153891231007802607925531582833121926445179311373588504552106016466983097049530752462731053682773354455513105149229148250528
14196880151946931209899519036497889352998557265916493552043969560076833946631425405698887172762025315453694094792941088854622079441009033910264186718892404227666488831015600315725452638294797441131799285562192129601590588419
Technical details
Node id
n-f550008e7c379093 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-35c14afe510ff622

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:      3,       5;
  2:     11,      37,       31;
  3:     45,     228,      384,       215;
  4:    197,    1334,     3380,      3798,      1597;
  5:    903,    7655,    25910,     43770,     36905,     12425;
  6:   4279,   43575,   184605,    416465,    527685,    356055,      99955;
  7:  20793,  247226,  1258068,   3551870,   6008730,   6090980,    3425730,    824675;
  8: 103049, 1401100,  8324344,  28227640,  59753764,  80858680,   68306728,  32935372,   6939769;
  9: 518859, 7940169, 53945964, 213569664, 542966004, 919297260, 1036556976, 750574404, 316711503, 59334605;
%F Formula
The row polynomials satisfy (n+2+(2*n+4)*x+(n+2)*x^2)*P(n,x) = -(n-10+(11*n-44)*x+(31*n-70)*x^2+(33*n-48)*x^3+(12*n-12)*x^4)*P(n-1,x) + (4+26*x+66*x^2+82*x^3+50*x^4+12*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[(-(n - 10 + (11*n - 44)*x + (31*n - 70)*x^2 + (33*n - 48)*x^3 + (12*n - 12)*x^4)*P[n - 1, x] + (4 + 26*x + 66*x^2 + 82*x^3 + 50*x^4 + 12*x^5)*D[P[n - 1, x], x])/(n + 2 + (2*n + 4)*x + (n + 2)*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] = (-(n - 10 + (11*n - 44)*x + (31*n - 70)*x^2 + (33*n - 48)*x^3 + (12*n - 12)*x^4)*v[i - 1] + (4 + 26*x + 66*x^2 + 82*x^3 + 50*x^4 + 12*x^5)*deriv(v[i - 1], x))/(n + 2 + (2*n + 4)*x + (n + 2)*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

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023