Real-rooted atlas

A001263 → basis A013611 → reverse

nonnegative coefficients

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

Recurrence

\(\left(2 + n\right)\,P_{n}(x) = \left(8 + 4\,x + 4\,n + 2\,n x\right)\,P_{n-1}(x) + \left(8\,x + 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1426
21620541
3641448414306
4256896896336422426
51024512076804800132013220076
64096276485760052800237605148429171481
7163841433603942404928003203201121122002014301500666

All 15 rows

Technical details
Node id
n-dcd53454e0276c00 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-c585b23acdf03520

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:      4,       2;
  2:     16,      20,        5;
  3:     64,     144,       84,       14;
  4:    256,     896,      896,      336,       42;
  5:   1024,    5120,     7680,     4800,     1320,      132;
  6:   4096,   27648,    57600,    52800,    23760,     5148,     429;
  7:  16384,  143360,   394240,   492800,   320320,   112112,   20020,    1430;
  8:  65536,  720896,  2523136,  4100096,  3587584,  1793792,  512512,   77792,   4862;
  9: 262144, 3538944, 15335424, 31309824, 35223552, 23482368, 9504768, 2291328, 302328, 16796;
%F Formula
The row polynomials satisfy (n+2)*P(n,x) = (4*n+8+(2*n+4)*x)*P(n-1,x) + (8*x+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[((4*n + 8 + (2*n + 4)*x)*P[n - 1, x] + (8*x + 2*x^2)*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; for(n = 1, N, my(i = n + 1); v[i] = ((4*n + 8 + (2*n + 4)*x)*v[i - 1] + (8*x + 2*x^2)*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

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