Real-rooted atlas

A132812 → basis A007318 → basis A007318

nonnegative coefficients

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

Recurrence

\(\left(2\,n + n^{2}\right)\,P_{n}(x) = \left(2 + 9\,n + 3\,n x + 7\,n^{2} + 3\,n^{2} x\right)\,P_{n-1}(x) + \left(-4 - 6\,x - 2\,x^{2} - 4\,n - 6\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1628
23321357
3180168484400
498512105209052810
55418825047401260150619824
62995354411391651410526252317140497
71663443510083037441383203528049283368999968

All 15 rows

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(\left(2\,n + n^{2}\right)\,P_{n}(x) = \left(-n - 2\,n x - n x^{2} - n^{2} - 2\,n^{2} x - n^{2} x^{2}\right)\,P_{n-2}(x) + \left(3 + x + 9\,n + 3\,n x + 6\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 2\,x\)
Technical details
Node id
n-2caa837a4ea68fa4 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-31df69d181e4ed73)
Fingerprint
f6-cb8a225300b7095e

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:       6,        2;
  2:      33,       21,        3;
  3:     180,      168,       48,        4;
  4:     985,     1210,      520,       90,       5;
  5:    5418,     8250,     4740,     1260,     150,       6;
  6:   29953,    54411,    39165,    14105,    2625,     231,      7;
  7:  166344,   351008,   303744,   138320,   35280,    4928,    336,     8;
  8:  927441,  2229156,  2253888,  1242360,  403956,   78120,   8568,   468,   9;
  9: 5188590, 13992930, 16189560, 10481520, 4150440, 1030680, 157920, 14040, 630, 10;
%F Formula
The row polynomials satisfy (n^2+2*n)*P(n,x) = (7*n^2+9*n+2+(3*n^2+3*n)*x)*P(n-1,x) - (4*n+4+(6*n+6)*x+(2*n+2)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((7*n^2 + 9*n + 2 + (3*n^2 + 3*n)*x)*P[n - 1, x] - (4*n + 4 + (6*n + 6)*x + (2*n + 2)*x^2)*D[P[n - 1, x], x])/(n^2 + 2*n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((7*n^2 + 9*n + 2 + (3*n^2 + 3*n)*x)*v[i - 1] - (4*n + 4 + (6*n + 6)*x + (2*n + 2)*x^2)*deriv(v[i - 1], x))/(n^2 + 2*n)); 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

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