Real-rooted atlas

A091042 → basis A007318 → D+1

nonnegative coefficients

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

Recurrence

\(\left(6 + 21\,x - 6\,x^{2} + 5\,n - 26\,n x + 4\,n x^{2} - 12\,n^{2} + 8\,n^{2} x + 4\,n^{3}\right)\,P_{n}(x) = \left(72 + 117\,x + 21\,x^{2} - 18\,x^{3} - 36\,n - 111\,n x - 50\,n x^{2} + 12\,n x^{3} - 32\,n^{2} + 4\,n^{2} x + 24\,n^{2} x^{2} + 16\,n^{3} + 12\,n^{3} x\right)\,P_{n-1}(x) + \left(18\,x + 30\,x^{2} + 4\,x^{3} - 8\,x^{4} - 16\,n x^{2} - 16\,n x^{3} - 8\,n^{2} x - 8\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
27310
33630571
4176224777484
5832144079215693229
638408448651221122751121198
717408465924659221632473244213137411
87782424576030336018560059040940866515881672
934406412533761845248141004859840014035217136952175609593
1015073286225920106618889805824524825616586243009602918413111935439314
1165536003027763259179008637665284139520016633344410009659558447124175021222549887
12283115521447034883180462083934126083010887681477181444661529692618241105104728642277231390338156
1312163481668157440016646144002326528000205455360011948851204630528001184768001944800019448001086802900258646824341
1452009369631708938248521777152132875550721331527680089765314564143661056130852454427801446438438400327412815724836272753564201494
152214592512145961779204281545523273710698496827040727046351237939234124267520128913899523395500032610341888721909765308160221676446629330652600955
Technical details
Node id
n-44217d0ff908a843 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-a13c061a726114e3

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:       7,       3;
   3:      36,      30,        5;
   4:     176,     224,       77,       7;
   5:     832,    1440,      792,     156,       9;
   6:    3840,    8448,     6512,    2112,     275,      11;
   7:   17408,   46592,    46592,   21632,    4732,     442,     13;
   8:   77824,  245760,   303360,  185600,   59040,    9408,    665,    15;
   9:  344064, 1253376,  1845248, 1410048,  598400,  140352,  17136,   952,   17;
  10: 1507328, 6225920, 10661888, 9805824, 5248256, 1658624, 300960, 29184, 1311, 19;
%F Formula
The row polynomials satisfy (4*n^3-12*n^2+5*n+6+(8*n^2-26*n+21)*x+(4*n-6)*x^2)*P(n,x) = (16*n^3-32*n^2-36*n+72+(12*n^3+4*n^2-111*n+117)*x+(24*n^2-50*n+21)*x^2+(12*n-18)*x^3)*P(n-1,x) - ((8*n^2-18)*x+(8*n^2+16*n-30)*x^2+(16*n-4)*x^3+8*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((16*n^3 - 32*n^2 - 36*n + 72 + (12*n^3 + 4*n^2 - 111*n + 117)*x + (24*n^2 - 50*n + 21)*x^2 + (12*n - 18)*x^3)*P[n - 1, x] - ((8*n^2 - 18)*x + (8*n^2 + 16*n - 30)*x^2 + (16*n - 4)*x^3 + 8*x^4)*D[P[n - 1, x], x])/(4*n^3 - 12*n^2 + 5*n + 6 + (8*n^2 - 26*n + 21)*x + (4*n - 6)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((16*n^3 - 32*n^2 - 36*n + 72 + (12*n^3 + 4*n^2 - 111*n + 117)*x + (24*n^2 - 50*n + 21)*x^2 + (12*n - 18)*x^3)*v[i - 1] - ((8*n^2 - 18)*x + (8*n^2 + 16*n - 30)*x^2 + (16*n - 4)*x^3 + 8*x^4)*deriv(v[i - 1], x))/(4*n^3 - 12*n^2 + 5*n + 6 + (8*n^2 - 26*n + 21)*x + (4*n - 6)*x^2)); 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