Real-rooted atlas

A104684 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(7\,x - 2\,x^{2} + 4\,n - 11\,n x + 2\,n x^{2} - 6\,n^{2} + 4\,n^{2} x + 2\,n^{3}\right)\,P_{n}(x) = \left(32 + 83\,x + 28\,x^{2} - 20\,x^{3} + 4\,n - 71\,n x - 56\,n x^{2} + 12\,n x^{3} - 38\,n^{2} - 8\,n^{2} x + 24\,n^{2} x^{2} + 14\,n^{3} + 12\,n^{3} x\right)\,P_{n-1}(x) + \left(4 + 18\,x + 22\,x^{2} - 8\,x^{4} + 2\,n - 2\,n x - 20\,n x^{2} - 16\,n x^{3} - 4\,n^{2} - 12\,n^{2} x - 8\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2527
32924457
4177234848503
5108920161200256164577
66693160501368049607203242135
7410051211761357807168018000192064389625
825033788092212291728632403197605980849281283608295
915232016225792104245129196544457632012759041863681228825633421185
1092398774304809884219840897550085652460821345408468787255296029952512309404135
1155897533292512600655375140820346880627735360301948416904512001615872015782407168010242862076793
12337339665195958362649501643407125725640643404960037798252801450428672355576320529267204364800168960204826450155671
13203145209712973455072364933847525943222912061946430480431206871042030162534464073687041315733760166256640117596163932164096244200780001
141220987594185039437522263675938032479535280224567241406160457364093472256106932224995813744642644309324846312780805043118083099033690521681922252364924919
15732603213575526939786481873109942340376402622668849851736814884574761935744297652420665613812256972804535477506561031317647361563059097614849802248013824020643841638420754653295801
Technical details
Node id
n-c994636ce6f2f2be · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-1c3940d3e6e6c9d9

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:       1;
   2:       5,        2;
   3:      29,       24,        4;
   4:     177,      234,       84,        8;
   5:    1089,     2016,     1200,      256,       16;
   6:    6693,    16050,    13680,     4960,      720,       32;
   7:   41005,   121176,   135780,    71680,    18000,     1920,      64;
   8:  250337,   880922,  1229172,   863240,   319760,    59808,    4928,    128;
   9: 1523201,  6225792, 10424512,  9196544,  4576320,  1275904,  186368,  12288,   256;
  10: 9239877, 43048098, 84219840, 89755008, 56524608, 21345408, 4687872, 552960, 29952, 512;
%F Formula
The row polynomials satisfy (2*n^3-6*n^2+4*n+(4*n^2-11*n+7)*x+(2*n-2)*x^2)*P(n,x) = (14*n^3-38*n^2+4*n+32+(12*n^3-8*n^2-71*n+83)*x+(24*n^2-56*n+28)*x^2+(12*n-20)*x^3)*P(n-1,x) - (4*n^2-2*n-4+(12*n^2+2*n-18)*x+(8*n^2+20*n-22)*x^2+16*n*x^3+8*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((14*n^3 - 38*n^2 + 4*n + 32 + (12*n^3 - 8*n^2 - 71*n + 83)*x + (24*n^2 - 56*n + 28)*x^2 + (12*n - 20)*x^3)*P[n - 1, x] - (4*n^2 - 2*n - 4 + (12*n^2 + 2*n - 18)*x + (8*n^2 + 20*n - 22)*x^2 + 16*n*x^3 + 8*x^4)*D[P[n - 1, x], x])/(2*n^3 - 6*n^2 + 4*n + (4*n^2 - 11*n + 7)*x + (2*n - 2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((14*n^3 - 38*n^2 + 4*n + 32 + (12*n^3 - 8*n^2 - 71*n + 83)*x + (24*n^2 - 56*n + 28)*x^2 + (12*n - 20)*x^3)*v[i - 1] - (4*n^2 - 2*n - 4 + (12*n^2 + 2*n - 18)*x + (8*n^2 + 20*n - 22)*x^2 + 16*n*x^3 + 8*x^4)*deriv(v[i - 1], x))/(2*n^3 - 6*n^2 + 4*n + (4*n^2 - 11*n + 7)*x + (2*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, v[i])))};
T(10)

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