Real-rooted atlas

A082985 → x ↦ x+1 → D

nonnegative coefficients

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

Recurrence

\(\left(2 - 6\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(-10 - 8\,x - 12\,n - 10\,n x + 16\,n^{2} + 12\,n^{2} x\right)\,P_{n-1}(x) + \left(-5 - 9\,x - 4\,x^{2} - 10\,n - 18\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
133
2151025
3567021147
418934219836765
560514301221440553751
618725460616233288457817745
756551962027540198607665147010581915
8167796749011332210234053210157082380136371365
9490962246184390714766723122651251722952636481711660239
101420657284901624938206136016305458280302671205191253552107340025
1140735323140305800071842048478093054807782197466552826486526759025332156323
12115920072258002010570032876800349869502533230012660200433180098257513805010450300139810125
133277611222426006803028012373020014862555012378474072963450304808408880300173745021235514040351603979767
1492165196764012222555794045166108060451370056943936038697078019117032068216265172100502944370316680184734062594876065
Technical details
Node id
n-93c87bd66e848d3f · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-2cb32fb53e87d6dc

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     0;
  1:     3;
  2:    15,     10;
  3:    56,     70,     21;
  4:   189,    342,    198,     36;
  5:   605,   1430,   1221,    440,     55;
  6:  1872,   5460,   6162,   3328,    845,     78;
  7:  5655,  19620,  27540,  19860,   7665,   1470,   105;
  8: 16779,  67490, 113322, 102340,  53210,  15708,  2380,  136;
  9: 49096, 224618, 439071, 476672, 312265, 125172, 29526, 3648, 171;
%F Formula
The row polynomials satisfy (4*n^2-6*n+2)*P(n,x) = (16*n^2-12*n-10+(12*n^2-10*n-8)*x)*P(n-1,x) - (10*n+5+(18*n+9)*x+(8*n+4)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((16*n^2 - 12*n - 10 + (12*n^2 - 10*n - 8)*x)*P[n - 1, x] - (10*n + 5 + (18*n + 9)*x + (8*n + 4)*x^2)*D[P[n - 1, x], x])/(4*n^2 - 6*n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((16*n^2 - 12*n - 10 + (12*n^2 - 10*n - 8)*x)*v[i - 1] - (10*n + 5 + (18*n + 9)*x + (8*n + 4)*x^2)*deriv(v[i - 1], x))/(4*n^2 - 6*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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023