Real-rooted atlas

A085478 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-66 + 24\,x + 24\,x^{2} + 116\,n - 16\,n x - 16\,n x^{2} - 72\,n^{2} + 16\,n^{3}\right)\,P_{n}(x) = \left(-4 + 68\,x + 64\,x^{2} + 16\,x^{3} + 30\,n - 44\,n x - 56\,n x^{2} - 16\,n x^{3} - 56\,n^{2} - 32\,n^{2} x + 24\,n^{3} + 16\,n^{3} x\right)\,P_{n-1}(x) + \left(-25 - 120\,x - 184\,x^{2} - 96\,x^{3} - 16\,x^{4} - 40\,n - 96\,n x - 32\,n x^{2} + 20\,n^{2} + 48\,n^{2} x + 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
011
111
2426
31518437
451102568217
5164466432152161230
6509188425081488384326805
715417044123241051245609286436973
8458024946542886100038000129602176128198078
913419848942214083102642546401245123494449922561049329
10388712802188527961435856147076894684838035290624112645125508109
1111155690288231429446189584762563261266243237248110233622809625088102428693014
1231768928527601118407225246656364231203520697623206400103841283066624560640552962048148506409
13898825886884038678520985200001631100001847152001464691208171136031680000825600013516801208324096764384473
142528836271990101306342083707743846934308809017058248377785605620869122715240969281536021638144320716826214481923915593718
Technical details
Node id
n-772e0ec3bbbc91e6 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-b9815478474a59b6

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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;
  2:     4,     2;
  3:    15,    18,      4;
  4:    51,   102,     56,      8;
  5:   164,   466,    432,    152,     16;
  6:   509,  1884,   2508,   1488,    384,     32;
  7:  1541,  7044,  12324,  10512,   4560,    928,    64;
  8:  4580, 24946,  54288,  61000,  38000,  12960,  2176,  128;
  9: 13419, 84894, 221408, 310264, 254640, 124512, 34944, 4992, 256;
%F Formula
The row polynomials satisfy (16*n^3-72*n^2+116*n-66-(16*n-24)*x-(16*n-24)*x^2)*P(n,x) = (24*n^3-56*n^2+30*n-4+(16*n^3-32*n^2-44*n+68)*x-(56*n-64)*x^2-(16*n-16)*x^3)*P(n-1,x) + (20*n^2-40*n-25+(48*n^2-96*n-120)*x+(16*n^2-32*n-184)*x^2-96*x^3-16*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[((24*n^3 - 56*n^2 + 30*n - 4 + (16*n^3 - 32*n^2 - 44*n + 68)*x - (56*n - 64)*x^2 - (16*n - 16)*x^3)*P[n - 1, x] + (20*n^2 - 40*n - 25 + (48*n^2 - 96*n - 120)*x + (16*n^2 - 32*n - 184)*x^2 - 96*x^3 - 16*x^4)*D[P[n - 1, x], x])/(16*n^3 - 72*n^2 + 116*n - 66 - (16*n - 24)*x - (16*n - 24)*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] = ((24*n^3 - 56*n^2 + 30*n - 4 + (16*n^3 - 32*n^2 - 44*n + 68)*x - (56*n - 64)*x^2 - (16*n - 16)*x^3)*v[i - 1] + (20*n^2 - 40*n - 25 + (48*n^2 - 96*n - 120)*x + (16*n^2 - 32*n - 184)*x^2 - 96*x^3 - 16*x^4)*deriv(v[i - 1], x))/(16*n^3 - 72*n^2 + 116*n - 66 - (16*n - 24)*x - (16*n - 24)*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

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