Real-rooted atlas

A094816 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-12 + 7\,n - n^{2}\right)\,P_{n-4}(x) + \left(9 - 6\,x - 9\,n + 2\,n x + 2\,n^{2}\right)\,P_{n-3}(x) + \left(4 + 2\,x - x^{2} - 2\,n x - n^{2}\right)\,P_{n-2}(x) + \left(-1 + 2\,x + 2\,n\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 3 + x\)
\(P_{3}(x) = 10 + 7\,x + x^{2}\)
\(P_{4}(x) = 39 + 41\,x + 12\,x^{2} + x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2314
3107118
4394112193
5181246107181553
699516099052252513760
76386116977879253541533128946
8470759473772933282245985700421249697
9392673851204726662323288827191254411065212390249
103659731842988978184263881282114653721031524066166263125171972
1137709394914009959077090149249650163073603464685480543431102400751289429114
124257795191078067737113424539866261823124086892557562637927095410098337309533558813609499773
1352288355251375252356215205174665945781916237173671269786786901777916692253576619834981184704565102148542832801
14693964044671887541267292179268987871430939908196011729495217170237942344593692349379801950666616368453818489960711171700453230880
1598989800266627743951060773328132597763229131332493710198190869233123132112106818217806410818764957125738369910676065465305242794617917133110796243234986
Technical details
Node id
n-8bc6801b2691aa61 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-e80bd10399b19c76

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:       3,       1;
   3:      10,       7,       1;
   4:      39,      41,      12,       1;
   5:     181,     246,     107,      18,       1;
   6:     995,    1609,     905,     225,      25,      1;
   7:    6386,   11697,    7879,    2535,     415,     33,     1;
   8:   47075,   94737,   72933,   28224,    5985,    700,    42,    1;
   9:  392673,  851204,  726662,  323288,   82719,  12544,  1106,   52,  1;
  10: 3659731, 8429889, 7818426, 3881282, 1146537, 210315, 24066, 1662, 63, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2-7*n+12)*P(n-4,x) + (2*n^2-9*n+9+(2*n-6)*x)*P(n-3,x) - (n^2-4+(2*n-2)*x+x^2)*P(n-2,x) + (2*n-1+2*x)*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 3 + x, P(3,x) = 10 + 7*x + x^2, P(4,x) = 39 + 41*x + 12*x^2 + x^3.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 3 + x;
P[3, x_] := 10 + 7*x + x^2;
P[4, x_] := 39 + 41*x + 12*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 7*n + 12)*P[n - 4, x] + (2*n^2 - 9*n + 9 + (2*n - 6)*x)*P[n - 3, x] - (n^2 - 4 + (2*n - 2)*x + x^2)*P[n - 2, x] + (2*n - 1 + 2*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + x; v[3] = 10 + 7*x + x^2; v[4] = 39 + 41*x + 12*x^2 + x^3; for(n = 5, N, my(i = n); v[i] = -(n^2 - 7*n + 12)*v[i - 4] + (2*n^2 - 9*n + 9 + (2*n - 6)*x)*v[i - 3] - (n^2 - 4 + (2*n - 2)*x + x^2)*v[i - 2] + (2*n - 1 + 2*x)*v[i - 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

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