Real-rooted atlas

A114656 → basis A007318 → reverse

nonnegative coefficients

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(1 - x - 4\,x^{2} + n + 5\,n x + 2\,n x^{2}\right)\,P_{n-1}(x) + \left(2\,x - 2\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2134
3181120
41155145112
5124146304197672
6135330119017659034224
71486453520883510128427927456
81631141871532515620135779920793183040

All 15 rows

Technical details
Node id
n-57c77e76c5c7eb67 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-c8aa5aaa0c2f2cd0

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 1,  3;
   3: 1,  8,   11;
   4: 1, 15,   51,    45;
   5: 1, 24,  146,   304,    197;
   6: 1, 35,  330,  1190,   1765,    903;
   7: 1, 48,  645,  3520,   8835,  10128,    4279;
   8: 1, 63, 1141,  8715,  32515,  62013,   57799,   20793;
   9: 1, 80, 1876, 19040,  98014, 274064,  419524,  329024,  103049;
  10: 1, 99, 2916, 37884, 256158, 977130, 2169300, 2766060, 1871145, 518859;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (n+1+(5*n-1)*x+(2*n-4)*x^2)*P(n-1,x) + (2*x-2*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n + 1 + (5*n - 1)*x + (2*n - 4)*x^2)*P[n - 1, x] + (2*x - 2*x^3)*D[P[n - 1, x], x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((n + 1 + (5*n - 1)*x + (2*n - 4)*x^2)*v[i - 1] + (2*x - 2*x^3)*deriv(v[i - 1], x))/(n + 1)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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