Real-rooted atlas

A001263 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 - 4\,x + 10\,n + 8\,n^{2} + 19\,n^{2} x + 4\,n^{2} x^{2} + 2\,n^{3} + 21\,n^{3} x + 10\,n^{3} x^{2} + 6\,n^{4} x + 8\,n^{4} x^{2} + 2\,n^{5} x^{2}\right)\,P_{n}(x) = \left(4 - 4\,x + 16\,n + 42\,n x + 8\,n x^{3} + 20\,n^{2} + 118\,n^{2} x + 116\,n^{2} x^{2} + 36\,n^{2} x^{3} + 8\,n^{3} + 96\,n^{3} x + 180\,n^{3} x^{2} + 80\,n^{3} x^{3} + 24\,n^{4} x + 80\,n^{4} x^{2} + 56\,n^{4} x^{3} + 8\,n^{5} x^{2} + 12\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(-4\,x - 14\,x^{2} - 16\,x^{3} - 8\,x^{4} - 8\,n x - 47\,n x^{2} - 80\,n x^{3} - 36\,n x^{4} - 4\,n^{2} x - 44\,n^{2} x^{2} - 104\,n^{2} x^{3} - 64\,n^{2} x^{4} - 12\,n^{3} x^{2} - 44\,n^{3} x^{3} - 40\,n^{3} x^{4} - 4\,n^{4} x^{3} - 8\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
12810
25403681
314168324128634
442672201617924004922
5132264010800153608000115238084
6429102965346010560090000311043136294025
714304004025225264064077000044352010976081922265834

All 15 rows

Technical details
Node id
n-0248f7b63cfca929 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-9d61137180e532ac

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:     2,      8;
  2:     5,     40,      36;
  3:    14,    168,     324,      128;
  4:    42,    672,    2016,     1792,      400;
  5:   132,   2640,   10800,    15360,     8000,     1152;
  6:   429,  10296,   53460,   105600,    90000,    31104,     3136;
  7:  1430,  40040,  252252,   640640,   770000,   443520,   109760,    8192;
  8:  4862, 155584, 1153152,  3587584,  5605600,  4612608,  1931776,  360448,   20736;
  9: 16796, 604656, 5155488, 19009536, 36691200, 39626496, 23971584, 7667712, 1119744, 51200;
%F Formula
The row polynomials satisfy (2*n^3+8*n^2+10*n+4+(6*n^4+21*n^3+19*n^2-4)*x+(2*n^5+8*n^4+10*n^3+4*n^2)*x^2)*P(n,x) = (8*n^3+20*n^2+16*n+4+(24*n^4+96*n^3+118*n^2+42*n-4)*x+(8*n^5+80*n^4+180*n^3+116*n^2)*x^2+(12*n^5+56*n^4+80*n^3+36*n^2+8*n)*x^3)*P(n-1,x) - ((4*n^2+8*n+4)*x+(12*n^3+44*n^2+47*n+14)*x^2+(4*n^4+44*n^3+104*n^2+80*n+16)*x^3+(8*n^4+40*n^3+64*n^2+36*n+8)*x^4)*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[((8*n^3 + 20*n^2 + 16*n + 4 + (24*n^4 + 96*n^3 + 118*n^2 + 42*n - 4)*x + (8*n^5 + 80*n^4 + 180*n^3 + 116*n^2)*x^2 + (12*n^5 + 56*n^4 + 80*n^3 + 36*n^2 + 8*n)*x^3)*P[n - 1, x] - ((4*n^2 + 8*n + 4)*x + (12*n^3 + 44*n^2 + 47*n + 14)*x^2 + (4*n^4 + 44*n^3 + 104*n^2 + 80*n + 16)*x^3 + (8*n^4 + 40*n^3 + 64*n^2 + 36*n + 8)*x^4)*D[P[n - 1, x], x])/(2*n^3 + 8*n^2 + 10*n + 4 + (6*n^4 + 21*n^3 + 19*n^2 - 4)*x + (2*n^5 + 8*n^4 + 10*n^3 + 4*n^2)*x^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] = ((8*n^3 + 20*n^2 + 16*n + 4 + (24*n^4 + 96*n^3 + 118*n^2 + 42*n - 4)*x + (8*n^5 + 80*n^4 + 180*n^3 + 116*n^2)*x^2 + (12*n^5 + 56*n^4 + 80*n^3 + 36*n^2 + 8*n)*x^3)*v[i - 1] - ((4*n^2 + 8*n + 4)*x + (12*n^3 + 44*n^2 + 47*n + 14)*x^2 + (4*n^4 + 44*n^3 + 104*n^2 + 80*n + 16)*x^3 + (8*n^4 + 40*n^3 + 64*n^2 + 36*n + 8)*x^4)*deriv(v[i - 1], x))/(2*n^3 + 8*n^2 + 10*n + 4 + (6*n^4 + 21*n^3 + 19*n^2 - 4)*x + (2*n^5 + 8*n^4 + 10*n^3 + 4*n^2)*x^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