Real-rooted atlas

A001263 → basis A013609 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1224
2410519
38364214100
41611222416842562
53232096012006601323304
664864360066005940257442920071
7128224012320308004004028028100101430124996

All 15 rows

Technical details
Node id
n-9e64d7bb1b0657f2 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-665cb75b242a18c2

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:   2,     2;
  2:   4,    10,      5;
  3:   8,    36,     42,     14;
  4:  16,   112,    224,    168,      42;
  5:  32,   320,    960,   1200,     660,     132;
  6:  64,   864,   3600,   6600,    5940,    2574,     429;
  7: 128,  2240,  12320,  30800,   40040,   28028,   10010,   1430;
  8: 256,  5632,  39424, 128128,  224224,  224224,  128128,  38896,   4862;
  9: 512, 13824, 119808, 489216, 1100736, 1467648, 1188096, 572832, 151164, 16796;
%F Formula
The row polynomials satisfy (n+2)*P(n,x) = (2*n+4+(2*n+4)*x)*P(n-1,x) + (4*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[((2*n + 4 + (2*n + 4)*x)*P[n - 1, x] + (4*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] = ((2*n + 4 + (2*n + 4)*x)*v[i - 1] + (4*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