Real-rooted atlas

A126216 → basis A007318 → reverse

nonnegative coefficients

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

Recurrence

\(\left(2 + n\right)\,P_{n}(x) = \left(2 + 8\,x + 4\,x^{2} + n + n x - 4\,n x^{2}\right)\,P_{n-1}(x) + \left(2\,x + 6\,x^{2} + 4\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1134
2171119
31124245100
4118104242197562
512521079013759033304
6133375201555957773427920071
7142616441017290379684387620793124996
8152952868044884138040250432247684103049793774
91631404157921030684150441048152161895613992935188595120632
10175199527030215880109006835847007671030103116457913955264672333463102
111882750440554207502584428106038242942115054585795649354204481287813648869221060008
12110236966897077368556521082808141696827148232114905379930650405273528254064726710393731473830308
13111748621043901355783115757076811318828335793284144789317744731292597702250251122552214421318993726935199904186192
1411336279153517228127922447425153768615755055015269712943570270530331322176755917502911517154687934318195232633196880151967015401391
Technical details
Node id
n-2a148c12d0f29911 · revision 0 · JSON
Definition source
derived:reversal
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: 1,  3;
  2: 1,  7,   11;
  3: 1, 12,   42,    45;
  4: 1, 18,  104,   242,    197;
  5: 1, 25,  210,   790,   1375,    903;
  6: 1, 33,  375,  2015,   5595,   7773,    4279;
  7: 1, 42,  616,  4410,  17290,  37968,   43876,   20793;
  8: 1, 52,  952,  8680,  44884, 138040,  250432,  247684,  103049;
  9: 1, 63, 1404, 15792, 103068, 415044, 1048152, 1618956, 1399293, 518859;
%F Formula
The row polynomials satisfy (n+2)*P(n,x) = (n+2+(n+8)*x-(4*n-4)*x^2)*P(n-1,x) + (2*x+6*x^2+4*x^3)*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[((n + 2 + (n + 8)*x - (4*n - 4)*x^2)*P[n - 1, x] + (2*x + 6*x^2 + 4*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n + 2 + (n + 8)*x - (4*n - 4)*x^2)*v[i - 1] + (2*x + 6*x^2 + 4*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

Candidate equalities ?

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