Real-rooted atlas

A126216 → D → D

nonnegative coefficients

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

Recurrence

\(\left(4 - 4\,n - n^{2} + n^{3}\right)\,P_{n}(x) = \left(-2\,n x - 2\,n^{2} - 5\,n^{2} x + 4\,n^{3} + 3\,n^{3} x\right)\,P_{n-1}(x) + \left(-2\,n^{2} x - 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12Pn(1)
00
10
222
318624
41128412208
5600720240201580
62970495027005403011190
7140143003023100770010504275936
8640641681681681688008018480184856500864
92864168910721100736687960229320393123024723237912
10125970045349206683040519792022932005733007644046809020623290
11547162022383900383724003581424019792080659736012936001386006930110129870840
1223535072107869080211048200230234400153489600644656321696464026928002376009900132810547056
13100440080509926560112183843214022980401097450640558702144186234048399072965250960388960137281565022451044
144259045702372896890580041462081800719007362064710441723882617925027124888643768729721096996906126121856418230937586862
Technical details
Node id
n-cdf8005c806e80e9 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-033fc5829f38f9ac

Observed on the stored rows ?

  • Degrees: -1, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 (affine: slope 1, intercept -2)
  • 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:      0;
  1:      0;
  2:      2;
  3:     18,      6;
  4:    112,     84,      12;
  5:    600,    720,     240,     20;
  6:   2970,   4950,    2700,    540,     30;
  7:  14014,  30030,   23100,   7700,   1050,    42;
  8:  64064, 168168,  168168,  80080,  18480,  1848,   56;
  9: 286416, 891072, 1100736, 687960, 229320, 39312, 3024, 72;
%F Formula
The row polynomials satisfy (n^3-n^2-4*n+4)*P(n,x) = (4*n^3-2*n^2+(3*n^3-5*n^2-2*n)*x)*P(n-1,x) - (2*n^2*x+2*n^2*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^3 - 2*n^2 + (3*n^3 - 5*n^2 - 2*n)*x)*P[n - 1, x] - (2*n^2*x + 2*n^2*x^2)*D[P[n - 1, x], x])/(n^3 - n^2 - 4*n + 4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 2; for(n = 3, N, my(i = n + 1); v[i] = ((4*n^3 - 2*n^2 + (3*n^3 - 5*n^2 - 2*n)*x)*v[i - 1] - (2*n^2*x + 2*n^2*x^2)*deriv(v[i - 1], x))/(n^3 - n^2 - 4*n + 4)); 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