Real-rooted atlas

A207823 → reverse → D+1

nonnegative coefficients

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

Recurrence

\(\left(6\,n + 10\,n x - 80\,n x^{2} + 24\,n x^{3} - 20\,n^{2} - 40\,n^{2} x + 48\,n^{2} x^{2} + 16\,n^{3} + 24\,n^{3} x\right)\,P_{n}(x) = \left(32\,x^{2} + 148\,x^{3} + 144\,x^{4} - 144\,x^{5} - 2\,n - 50\,n x - 192\,n x^{2} - 348\,n x^{3} - 336\,n x^{4} + 144\,n x^{5} - 4\,n^{2} - 92\,n^{2} x - 264\,n^{2} x^{2} + 48\,n^{2} x^{3} + 288\,n^{2} x^{4} + 16\,n^{3} + 120\,n^{3} x + 240\,n^{3} x^{2} + 144\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(3\,x^{2} + 28\,x^{3} + 56\,x^{4} - 48\,x^{5} - 144\,x^{6} - 2\,n x - 20\,n x^{2} - 80\,n x^{3} - 240\,n x^{4} - 288\,n x^{5} - 8\,n^{2} x - 76\,n^{2} x^{2} - 192\,n^{2} x^{3} - 144\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1549
29381562
31310421456387
41720278910682092285
52133219324956499178013012
6254943835146642792622378291172233
729688669034160969301465929751610864393469
8339141068968436259375584052732006416184405452112234
937117216024123508587209176586832926683521976174821715131611207995
104114622288720641611820764459854111024131765408816465959725217456471958912089
11451784314703252242180436991132830953832656874809101207175241224297814422107560307233896
12492138419654890203758685200084887529146820041582437043479045462683033746353912128195678655211591680273
13532524545647079166138275374723641651579925296941601227811734200938116022128851601490281140490458867293545248199400433
145729426945999304895908346607134633401897112548116323504185154719021197210554936770105402952806495728700197159509010955257542032063830
Technical details
Node id
n-f40d2afe5666e904 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-5a664c14c82b88c9

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:  5,    4;
  2:  9,   38,    15;
  3: 13,  104,   214,     56;
  4: 17,  202,   789,   1068,    209;
  5: 21,  332,  1932,   4956,   4991,     780;
  6: 25,  494,  3835,  14664,  27926,   22378,    2911;
  7: 29,  688,  6690,  34160,  96930,  146592,   97516,   10864;
  8: 33,  914, 10689,  68436, 259375,  584052,  732006,  416184,   40545;
  9: 37, 1172, 16024, 123508, 587209, 1765868, 3292668, 3521976, 1748217, 151316;
%F Formula
The row polynomials satisfy (16*n^3-20*n^2+6*n+(24*n^3-40*n^2+10*n)*x+(48*n^2-80*n)*x^2+24*n*x^3)*P(n,x) = (16*n^3-4*n^2-2*n+(120*n^3-92*n^2-50*n)*x+(240*n^3-264*n^2-192*n+32)*x^2+(144*n^3+48*n^2-348*n+148)*x^3+(288*n^2-336*n+144)*x^4+(144*n-144)*x^5)*P(n-1,x) - ((8*n^2+2*n)*x+(76*n^2+20*n-3)*x^2+(192*n^2+80*n-28)*x^3+(144*n^2+240*n-56)*x^4+(288*n+48)*x^5+144*x^6)*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[((16*n^3 - 4*n^2 - 2*n + (120*n^3 - 92*n^2 - 50*n)*x + (240*n^3 - 264*n^2 - 192*n + 32)*x^2 + (144*n^3 + 48*n^2 - 348*n + 148)*x^3 + (288*n^2 - 336*n + 144)*x^4 + (144*n - 144)*x^5)*P[n - 1, x] - ((8*n^2 + 2*n)*x + (76*n^2 + 20*n - 3)*x^2 + (192*n^2 + 80*n - 28)*x^3 + (144*n^2 + 240*n - 56)*x^4 + (288*n + 48)*x^5 + 144*x^6)*D[P[n - 1, x], x])/(16*n^3 - 20*n^2 + 6*n + (24*n^3 - 40*n^2 + 10*n)*x + (48*n^2 - 80*n)*x^2 + 24*n*x^3)]];
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] = ((16*n^3 - 4*n^2 - 2*n + (120*n^3 - 92*n^2 - 50*n)*x + (240*n^3 - 264*n^2 - 192*n + 32)*x^2 + (144*n^3 + 48*n^2 - 348*n + 148)*x^3 + (288*n^2 - 336*n + 144)*x^4 + (144*n - 144)*x^5)*v[i - 1] - ((8*n^2 + 2*n)*x + (76*n^2 + 20*n - 3)*x^2 + (192*n^2 + 80*n - 28)*x^3 + (144*n^2 + 240*n - 56)*x^4 + (288*n + 48)*x^5 + 144*x^6)*deriv(v[i - 1], x))/(16*n^3 - 20*n^2 + 6*n + (24*n^3 - 40*n^2 + 10*n)*x + (48*n^2 - 80*n)*x^2 + 24*n*x^3)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023