Real-rooted atlas

A111125 → D → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(24 + 12\,x + 24\,x^{2} - 28\,n - 50\,n x - 100\,n x^{2} + 8\,n^{2} + 58\,n^{2} x + 170\,n^{2} x^{2} - 26\,n^{3} x - 151\,n^{3} x^{2} + 4\,n^{4} x + 74\,n^{4} x^{2} - 19\,n^{5} x^{2} + 2\,n^{6} x^{2}\right)\,P_{n}(x) = \left(-4\,n - 18\,n x - 6\,n x^{2} - n x^{3} + 8\,n^{2} + 26\,n^{2} x + 26\,n^{2} x^{2} + 5\,n^{2} x^{3} - 14\,n^{3} x - 43\,n^{3} x^{2} - 10\,n^{3} x^{3} + 4\,n^{4} x + 34\,n^{4} x^{2} + 10\,n^{4} x^{3} - 13\,n^{5} x^{2} - 5\,n^{5} x^{3} + 2\,n^{6} x^{2} + n^{6} x^{3}\right)\,P_{n-1}(x) + \left(-16\,x - 12\,x^{2} - 6\,x^{3} - x^{4} + 16\,n x + 28\,n x^{2} + 26\,n x^{3} + 5\,n x^{4} - 24\,n^{2} x^{2} - 46\,n^{2} x^{3} - 10\,n^{2} x^{4} + 8\,n^{3} x^{2} + 42\,n^{3} x^{3} + 10\,n^{3} x^{4} - 20\,n^{4} x^{3} - 5\,n^{4} x^{4} + 4\,n^{5} x^{3} + n^{5} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
10
211
35813
414562797
53021624364553
65561611887041252688
791145642124160162521611760
81403024121501760011250324034347747
92045712302945984055250257045831512183347
102851003267716173888216125143640521369728729674279
1138516632138996448448716625635040326536967681530910002395739
125062631226640910524802093000236476816093566712321676702300013318276064
136504004048262522880005525000771120066473403648000127575027500033275172827928608
14819589688339764667520134257502260483223930424165473287577955227700043124446656219792404669
15101584448138434490238723044275060698160771091446516787237440711146740003859900651456637132744300604129
Technical details
Node id
n-f8bc4c6c2916f399 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-fb58a652871ce32f

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -2)
  • 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:   0;
   2:   1;
   3:   5,     8;
   4:  14,    56,    27;
   5:  30,   216,   243,     64;
   6:  55,   616,  1188,    704,    125;
   7:  91,  1456,  4212,   4160,   1625,    216;
   8: 140,  3024, 12150,  17600,  11250,   3240,   343;
   9: 204,  5712, 30294,  59840,  55250,  25704,  5831,  512;
  10: 285, 10032, 67716, 173888, 216125, 143640, 52136, 9728, 729;
%F Formula
The row polynomials satisfy (8*n^2-28*n+24+(4*n^4-26*n^3+58*n^2-50*n+12)*x+(2*n^6-19*n^5+74*n^4-151*n^3+170*n^2-100*n+24)*x^2)*P(n,x) = (8*n^2-4*n+(4*n^4-14*n^3+26*n^2-18*n)*x+(2*n^6-13*n^5+34*n^4-43*n^3+26*n^2-6*n)*x^2+(n^6-5*n^5+10*n^4-10*n^3+5*n^2-n)*x^3)*P(n-1,x) + ((16*n-16)*x+(8*n^3-24*n^2+28*n-12)*x^2+(4*n^5-20*n^4+42*n^3-46*n^2+26*n-6)*x^3+(n^5-5*n^4+10*n^3-10*n^2+5*n-1)*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 4*n + (4*n^4 - 14*n^3 + 26*n^2 - 18*n)*x + (2*n^6 - 13*n^5 + 34*n^4 - 43*n^3 + 26*n^2 - 6*n)*x^2 + (n^6 - 5*n^5 + 10*n^4 - 10*n^3 + 5*n^2 - n)*x^3)*P[n - 1, x] + ((16*n - 16)*x + (8*n^3 - 24*n^2 + 28*n - 12)*x^2 + (4*n^5 - 20*n^4 + 42*n^3 - 46*n^2 + 26*n - 6)*x^3 + (n^5 - 5*n^4 + 10*n^3 - 10*n^2 + 5*n - 1)*x^4)*D[P[n - 1, x], x])/(8*n^2 - 28*n + 24 + (4*n^4 - 26*n^3 + 58*n^2 - 50*n + 12)*x + (2*n^6 - 19*n^5 + 74*n^4 - 151*n^3 + 170*n^2 - 100*n + 24)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((8*n^2 - 4*n + (4*n^4 - 14*n^3 + 26*n^2 - 18*n)*x + (2*n^6 - 13*n^5 + 34*n^4 - 43*n^3 + 26*n^2 - 6*n)*x^2 + (n^6 - 5*n^5 + 10*n^4 - 10*n^3 + 5*n^2 - n)*x^3)*v[i - 1] + ((16*n - 16)*x + (8*n^3 - 24*n^2 + 28*n - 12)*x^2 + (4*n^5 - 20*n^4 + 42*n^3 - 46*n^2 + 26*n - 6)*x^3 + (n^5 - 5*n^4 + 10*n^3 - 10*n^2 + 5*n - 1)*x^4)*deriv(v[i - 1], x))/(8*n^2 - 28*n + 24 + (4*n^4 - 26*n^3 + 58*n^2 - 50*n + 12)*x + (2*n^6 - 19*n^5 + 74*n^4 - 151*n^3 + 170*n^2 - 100*n + 24)*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

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