Real-rooted atlas

A084534 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(2 + 6\,x + 2\,x^{2} - n - 3\,n x - n x^{2}\right)\,P_{n-2}(x) + \left(6\,x + 4\,x^{2} - 3\,n x - 2\,n x^{2}\right)\,P_{n-2}'(x) + \left(2\,x^{2} - n x^{2}\right)\,P_{n-2}''(x) + \left(-6 - 4\,x + 3\,n + 2\,n x\right)\,P_{n-1}(x) + \left(-4\,x + 2\,n x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 2\,x\)
\(P_{2}(x) = 7 + 10\,x + 2\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1325
2710219
3184721288
447210166362461
512390211404305522652
632237757193420292578216497
78431554243008364421219417571052109893
82207633302482062926361388823004430521362778495
9577825636713984772230466143713843450865688495617125833551
1015127103345077493621638820013915470562095811753341311007635210246036848
113960341547024243852811725936212844544967222122185674722842356243474112752532381224598
121036821667353523045246582272113011442887517589656262683770345381362862882494265581608230023302127042
1327144366837422124375446056880209109923709510820497494736314475999239407581404950691293835471214022282351229837125247
14710647267724810668241252238886318240842946850208577943340446720347324147411947562824730948336749413250640391141686301214062280560543338
Technical details
Node id
n-7bc19a61beac35de · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-057772c6c1047f32

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,      2;
  2:    7,     10,       2;
  3:   18,     47,      21,       2;
  4:   47,    210,     166,      36,       2;
  5:  123,    902,    1140,     430,      55,      2;
  6:  322,   3775,    7193,    4202,     925,     78,     2;
  7:  843,  15542,   43008,   36442,   12194,   1757,   105,    2;
  8: 2207,  63330,  248206,  292636,  138882,  30044,  3052,  136,   2;
  9: 5778, 256367, 1398477, 2230466, 1437138, 434508, 65688, 4956, 171, 2;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = -(n-2+(3*n-6)*x+(n-2)*x^2)*P(n-2,x) - ((3*n-6)*x+(2*n-4)*x^2)*d/dx P(n-2,x) - (n-2)*x^2*d^2/dx^2 P(n-2,x) + (3*n-6+(2*n-4)*x)*P(n-1,x) + (2*n-4)*x*d/dx P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 2*x, P(2,x) = 7 + 10*x + 2*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[2, x_] := 7 + 10*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 2 + (3*n - 6)*x + (n - 2)*x^2)*P[n - 2, x] - ((3*n - 6)*x + (2*n - 4)*x^2)*D[P[n - 2, x], x] - (n - 2)*x^2*D[P[n - 2, x], {x, 2}] + (3*n - 6 + (2*n - 4)*x)*P[n - 1, x] + (2*n - 4)*x*D[P[n - 1, x], x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 2*x; v[3] = 7 + 10*x + 2*x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(n - 2 + (3*n - 6)*x + (n - 2)*x^2)*v[i - 2] - ((3*n - 6)*x + (2*n - 4)*x^2)*deriv(v[i - 2], x) - (n - 2)*x^2*deriv(deriv(v[i - 2], x), x) + (3*n - 6 + (2*n - 4)*x)*v[i - 1] + (2*n - 4)*x*deriv(v[i - 1], x))/(n - 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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023