Real-rooted atlas

A197208 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(5 + 6\,n + n^{2}\right)\,P_{n}(x) = \left(40 + 20\,x + 39\,n + 30\,n x + 11\,n^{2} + 10\,n^{2} x\right)\,P_{n-1}(x) + \left(12 - 3\,x - 15\,x^{2} - 12\,n - 27\,n x - 15\,n x^{2}\right)\,P_{n-1}'(x) + \left(12 + 30\,x + 24\,x^{2} + 6\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 180 + 180\,x + 30\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
118018030390
213201980780604140
396601932012180252010543785
470896177240154560546006720168464184
552264815679441750140887040197820156242524941468
6387144013550040184539601225980040647606123603276036052845480
7288099901152399601853775001527926406797637015744960167706063360495567682335
8215318400968932800179835480017725554009947322003162852005341644041619601148406606123872700

All 15 rows

Technical details
Node id
n-ed5623c8c46569fa · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-bb0a136ce054ce63

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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:         180,         180,           30;
   2:        1320,        1980,          780,           60;
   3:        9660,       19320,        12180,         2520,          105;
   4:       70896,      177240,       154560,        54600,         6720,         168;
   5:      522648,     1567944,      1750140,       887040,       197820,       15624,         252;
   6:     3871440,    13550040,     18453960,     12259800,      4064760,      612360,       32760,        360;
   7:    28809990,   115239960,    185377500,    152792640,     67976370,    15744960,     1677060,      63360,       495;
   8:   215318400,   968932800,   1798354800,   1772555400,    994732200,   316285200,    53416440,    4161960,    114840,      660;
   9:  1615579680,  8077898400,  16995092400,  19512979200,  13257644400,  5404679280,  1280539260,  162625320,   9523800,   197340,    858;
  10: 12165491520, 66910203360, 157382665440, 206395469280, 164915150400, 82395593280, 25429884480, 4624139520, 452251800, 20360340, 324324, 1092;
%F Formula
The row polynomials satisfy (n^2+6*n+5)*P(n,x) = (11*n^2+39*n+40+(10*n^2+30*n+20)*x)*P(n-1,x) - (12*n-12+(27*n+3)*x+(15*n+15)*x^2)*d/dx P(n-1,x) + (12+30*x+24*x^2+6*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 180 + 180*x + 30*x^2.
%t Mathematica
P[1, x_] := 180 + 180*x + 30*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((11*n^2 + 39*n + 40 + (10*n^2 + 30*n + 20)*x)*P[n - 1, x] - (12*n - 12 + (27*n + 3)*x + (15*n + 15)*x^2)*D[P[n - 1, x], x] + (12 + 30*x + 24*x^2 + 6*x^3)*D[P[n - 1, x], {x, 2}])/(n^2 + 6*n + 5)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 180 + 180*x + 30*x^2; for(n = 2, N, my(i = n); v[i] = ((11*n^2 + 39*n + 40 + (10*n^2 + 30*n + 20)*x)*v[i - 1] - (12*n - 12 + (27*n + 3)*x + (15*n + 15)*x^2)*deriv(v[i - 1], x) + (12 + 30*x + 24*x^2 + 6*x^3)*deriv(deriv(v[i - 1], x), x))/(n^2 + 6*n + 5)); 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