Real-rooted atlas

A069466 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(n^{2} + 2\,n^{2} x + n^{2} x^{2}\right)\,P_{n}(x) = \left(4 + 20\,x + 36\,x^{2} + 28\,x^{3} + 8\,x^{4} - 16\,n - 74\,n x - 124\,n x^{2} - 90\,n x^{3} - 24\,n x^{4} + 16\,n^{2} + 68\,n^{2} x + 104\,n^{2} x^{2} + 68\,n^{2} x^{3} + 16\,n^{2} x^{4}\right)\,P_{n-1}(x) + \left(4\,x + 20\,x^{2} + 36\,x^{3} + 28\,x^{4} + 8\,x^{5} - 8\,n x - 40\,n x^{2} - 72\,n x^{3} - 56\,n x^{4} - 16\,n x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
14610
23610878222
34001800264012606100
4490029400651006300022470184870
5635044762801411200206388014893204241165928300
68537767683984285238805588352060928560350436248305836197223180
7117786241236755525517832321355674320198053856017202144968223895681669290486732983400
816563690019876428001035782748030609699120560999439006528642120047108061000192676255203419932230234302790150
92363904400319127094001902247776006570263700001448962154640211562491140020449805376001261779433200450921814200711098130608274906425500
10341347795365120216930403435935045400135820491206403502038155616061538666677488746291432686806167087289792033232817372040105448544744401496053026468295696929911812
1149763430662482109660592966125641345824027272137268376080504509333248016542935828939522414583282279168250313715307656018061159553236808638128767268002464549691123523177739707760810667906696330520
12731245967233613162427410204810806543663595685350812881974560177948251936254804187123492180428871474490835011264891780059016887048071191097593624051675638631336720222155207097904085757752675931984680354749147164387930138576380764
131081724803600002109363367020000188998957684992001030213068452568003811344469852920001010540962268838000197545416763094400028826596458275616003139903893620894400252131093068654800014507004730458240005663657676631500001344710827139316001466415559777140014201344264501891000
14160934159556000033796173506760000328156672383540000195266780261280000079545177385770096002346618531789932400051696349457923170000863949985824594240001100598567316007760001063535800441081968007673742056423242800040088582771956992000143337736685510640003139751333106234000317877850826299800522859124051339578200
Technical details
Node id
n-a0d580d2a8ee89f3 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-c277ed2a39f7c92c

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,           6;
  2:         36,         108,           78;
  3:        400,        1800,         2640,         1260;
  4:       4900,       29400,        65100,        63000,         22470;
  5:      63504,      476280,      1411200,      2063880,       1489320,        424116;
  6:     853776,     7683984,     28523880,     55883520,      60928560,      35043624,       8305836;
  7:   11778624,   123675552,    551783232,   1355674320,    1980538560,    1720214496,     822389568,     166929048;
  8:  165636900,  1987642800,  10357827480,  30609699120,   56099943900,   65286421200,   47108061000,   19267625520,   3419932230;
  9: 2363904400, 31912709400, 190224777600, 657026370000, 1448962154640, 2115624911400, 2044980537600, 1261779433200, 450921814200, 71109813060;
%F Formula
The row polynomials satisfy (n^2+2*n^2*x+n^2*x^2)*P(n,x) = (16*n^2-16*n+4+(68*n^2-74*n+20)*x+(104*n^2-124*n+36)*x^2+(68*n^2-90*n+28)*x^3+(16*n^2-24*n+8)*x^4)*P(n-1,x) - ((8*n-4)*x+(40*n-20)*x^2+(72*n-36)*x^3+(56*n-28)*x^4+(16*n-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[((16*n^2 - 16*n + 4 + (68*n^2 - 74*n + 20)*x + (104*n^2 - 124*n + 36)*x^2 + (68*n^2 - 90*n + 28)*x^3 + (16*n^2 - 24*n + 8)*x^4)*P[n - 1, x] - ((8*n - 4)*x + (40*n - 20)*x^2 + (72*n - 36)*x^3 + (56*n - 28)*x^4 + (16*n - 8)*x^5)*D[P[n - 1, x], x])/(n^2 + 2*n^2*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] = ((16*n^2 - 16*n + 4 + (68*n^2 - 74*n + 20)*x + (104*n^2 - 124*n + 36)*x^2 + (68*n^2 - 90*n + 28)*x^3 + (16*n^2 - 24*n + 8)*x^4)*v[i - 1] - ((8*n - 4)*x + (40*n - 20)*x^2 + (72*n - 36)*x^3 + (56*n - 28)*x^4 + (16*n - 8)*x^5)*deriv(v[i - 1], x))/(n^2 + 2*n^2*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

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