Real-rooted atlas

A133214 → basis A007318 → reverse

nonnegative coefficients

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

Recurrence

\(\left(2 + n\right)\,P_{n}(x) = \left(4 + 4\,x - 2\,x^{2} + 2\,n + n x - 2\,n x^{2}\right)\,P_{n-1}(x) + \left(4\,x + 6\,x^{2} + 2\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 4 + 16\,x + 13\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
04161333
186011463245
2161926247683211921
332560272055205010168315525
4641536103203040044580320168989127905
5128403235616141680296520339612201698486391067925
62561024011468858777616374402646784248416012574722657299004545
751225344350208223641679269121689004822194816176353927777314146256376499525

All 15 rows

Technical details
Node id
n-fa10d6b5292a8c39 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-e7d8403424fc1d9b

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 2)
  • 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:    4,     16,      13;
  1:    8,     60,     114,       63;
  2:   16,    192,     624,      768,       321;
  3:   32,    560,    2720,     5520,      5010,      1683;
  4:   64,   1536,   10320,    30400,     44580,     32016,      8989;
  5:  128,   4032,   35616,   141680,    296520,    339612,    201698,      48639;
  6:  256,  10240,  114688,   587776,   1637440,   2646784,   2484160,    1257472,     265729;
  7:  512,  25344,  350208,  2236416,   7926912,  16890048,  22194816,   17635392,    7777314,   1462563;
  8: 1024,  61440, 1025280,  7956480,  34755840,  93413376, 160668480,  177672960,  122356260,  47800080,   8097453;
  9: 2048, 146432, 2900480, 26822400, 140997120, 463448832, 999132288, 1438388160, 1372173000, 833645340, 292292946, 45046719;
%F Formula
The row polynomials satisfy (n+2)*P(n,x) = (2*n+4+(n+4)*x-(2*n+2)*x^2)*P(n-1,x) + (4*x+6*x^2+2*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 4 + 16*x + 13*x^2.
%t Mathematica
P[0, x_] := 4 + 16*x + 13*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 4 + (n + 4)*x - (2*n + 2)*x^2)*P[n - 1, x] + (4*x + 6*x^2 + 2*x^3)*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] = 4 + 16*x + 13*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + 4 + (n + 4)*x - (2*n + 2)*x^2)*v[i - 1] + (4*x + 6*x^2 + 2*x^3)*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