Real-rooted atlas

A104684 → basis A013611 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-2 + 6\,x - 8\,x^{2} + 2\,n - 14\,n x + 24\,n x^{2} + 8\,n^{2} x - 24\,n^{2} x^{2} + 8\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-22 + 18\,x + 152\,x^{2} + 64\,x^{3} + 14\,n - 114\,n x - 232\,n x^{2} + 64\,n x^{3} + 64\,n^{2} x - 8\,n^{2} x^{2} - 224\,n^{2} x^{3} + 56\,n^{3} x^{2} + 96\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-1 + 32\,x^{2} + 72\,x^{3} + 32\,x^{4} - 8\,n x - 44\,n x^{2} - 40\,n x^{3} + 32\,n x^{4} - 8\,n^{2} x^{2} - 48\,n^{2} x^{3} - 64\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
23811
3136448125
4634567202561495
5321307274886144128018305
61683200406624087040448006144225947
78989128064534960972800825600294912286722793997
84863980679240753449488640113344006838272180633613107234529495

All 15 rows

Technical details
Node id
n-0f84e233ec4cc845 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-f9fc0a64807f736f

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:       3,        8;
   3:      13,       64,        48;
   4:      63,      456,       720,       256;
   5:     321,     3072,      7488,      6144,       1280;
   6:    1683,    20040,     66240,     87040,      44800,       6144;
   7:    8989,   128064,    534960,    972800,     825600,     294912,      28672;
   8:   48639,   806792,   4075344,   9488640,   11334400,    6838272,    1806336,    131072;
   9:  265729,  5029888,  29809920,  84697088,  130995200,  112852992,   51380224,  10485760,   589824;
  10: 1462563, 31109256, 211624704, 710234112, 1351203840, 1521967104, 1001914368, 358612992, 58392576, 2621440;
%F Formula
The row polynomials satisfy (2*n-2+(8*n^2-14*n+6)*x+(8*n^3-24*n^2+24*n-8)*x^2)*P(n,x) = (14*n-22+(64*n^2-114*n+18)*x+(56*n^3-8*n^2-232*n+152)*x^2+(96*n^3-224*n^2+64*n+64)*x^3)*P(n-1,x) - (1+8*n*x+(8*n^2+44*n-32)*x^2+(48*n^2+40*n-72)*x^3+(64*n^2-32*n-32)*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 - 22 + (64*n^2 - 114*n + 18)*x + (56*n^3 - 8*n^2 - 232*n + 152)*x^2 + (96*n^3 - 224*n^2 + 64*n + 64)*x^3)*P[n - 1, x] - (1 + 8*n*x + (8*n^2 + 44*n - 32)*x^2 + (48*n^2 + 40*n - 72)*x^3 + (64*n^2 - 32*n - 32)*x^4)*D[P[n - 1, x], x])/(2*n - 2 + (8*n^2 - 14*n + 6)*x + (8*n^3 - 24*n^2 + 24*n - 8)*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 - 22 + (64*n^2 - 114*n + 18)*x + (56*n^3 - 8*n^2 - 232*n + 152)*x^2 + (96*n^3 - 224*n^2 + 64*n + 64)*x^3)*v[i - 1] - (1 + 8*n*x + (8*n^2 + 44*n - 32)*x^2 + (48*n^2 + 40*n - 72)*x^3 + (64*n^2 - 32*n - 32)*x^4)*deriv(v[i - 1], x))/(2*n - 2 + (8*n^2 - 14*n + 6)*x + (8*n^3 - 24*n^2 + 24*n - 8)*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